{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 第2章 预备知识"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2.1 数据操作"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.1.1\n",
    "\n",
    "运行本节中的代码。将本节中的条件语句`X == Y`更改为`X < Y`或`X > Y`，然后看看你可以得到什么样的张量。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：** "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;将`X == Y`语句改为`X < Y`或`X > Y`，运行代码后，会返回和 X 大小相同的张量，对于每个位置，如果 X 的当前位置元素小于（大于）Y 的当前位置元素，则新张量中相应项的值为`True`， 反之为`False`。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "import torch\n",
    "import os\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from torch.distributions import multinomial\n",
    "from d2l import torch as d2l"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = torch.arange(12, dtype=torch.float32).reshape((3,4))\n",
    "Y = torch.tensor([[2.0, 1, 4, 3], [1, 2, 3, 4], [4, 3, 2, 1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[ True, False,  True, False],\n",
       "        [False, False, False, False],\n",
       "        [False, False, False, False]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X < Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[False, False, False, False],\n",
       "        [ True,  True,  True,  True],\n",
       "        [ True,  True,  True,  True]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X > Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.1.2\n",
    "\n",
    "用其他形状（例如三维张量）替换广播机制中按元素操作的两个张量。结果是否与预期相同？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：** "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- 能否得到预期结果要看两个张量的shape是否满足广播机制\n",
    "- 广播机制条件\n",
    "    1. 每个张量至少有一个维度\n",
    "    2. 广播时, 从右往左对比两个张量的shape, 需满足shape一样且大小匹配或shape不一样且大小不匹配的位置任意有一个张量为1或shape不一样且不匹配位置任意有一个张量不存在"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "二维情况\n",
      "tensor([[0],\n",
      "        [1],\n",
      "        [2]]) tensor([[0, 1]]) tensor([[0, 1],\n",
      "        [1, 2],\n",
      "        [2, 3]])\n"
     ]
    }
   ],
   "source": [
    "#二维情况\n",
    "print('二维情况')\n",
    "a = torch.arange(3).reshape((3, 1))\n",
    "b = torch.arange(2).reshape((1, 2))\n",
    "print(a, b, a+b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "三维情况\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(tensor([[[0],\n",
       "          [1]],\n",
       " \n",
       "         [[2],\n",
       "          [3]],\n",
       " \n",
       "         [[4],\n",
       "          [5]]]),\n",
       " tensor([[[0],\n",
       "          [1]],\n",
       " \n",
       "         [[2],\n",
       "          [3]],\n",
       " \n",
       "         [[4],\n",
       "          [5]]]),\n",
       " tensor([[[ 0],\n",
       "          [ 2]],\n",
       " \n",
       "         [[ 4],\n",
       "          [ 6]],\n",
       " \n",
       "         [[ 8],\n",
       "          [10]]]))"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#三维情况\n",
    "print('三维情况')\n",
    "# shape一样且大小匹配\n",
    "a = torch.arange(6).reshape((3,2,1))\n",
    "b = torch.arange(6).reshape((3,2,1))\n",
    "a, b, a+b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor([[[0],\n",
       "          [1],\n",
       "          [2]]]),\n",
       " tensor([[[0, 1, 2]],\n",
       " \n",
       "         [[3, 4, 5]],\n",
       " \n",
       "         [[6, 7, 8]]]),\n",
       " tensor([[[ 0,  1,  2],\n",
       "          [ 1,  2,  3],\n",
       "          [ 2,  3,  4]],\n",
       " \n",
       "         [[ 3,  4,  5],\n",
       "          [ 4,  5,  6],\n",
       "          [ 5,  6,  7]],\n",
       " \n",
       "         [[ 6,  7,  8],\n",
       "          [ 7,  8,  9],\n",
       "          [ 8,  9, 10]]]))"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# shape不一样且大小不匹配的位置任意有一个张量为1\n",
    "a = torch.arange(3).reshape((1,3,1))\n",
    "b = torch.arange(9).reshape((3,1,3))\n",
    "a, b, a+b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor([[0, 1, 2]]),\n",
       " tensor([[[0, 1, 2]],\n",
       " \n",
       "         [[3, 4, 5]],\n",
       " \n",
       "         [[6, 7, 8]]]),\n",
       " tensor([[[ 0,  2,  4]],\n",
       " \n",
       "         [[ 3,  5,  7]],\n",
       " \n",
       "         [[ 6,  8, 10]]]))"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# shape不一样且不匹配位置任意有一个张量不存在\n",
    "a = torch.arange(3).reshape((1,3))\n",
    "b = torch.arange(9).reshape((3,1,3))\n",
    "a, b, a+b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2.2 数据预处理"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "创建包含更多行和列的原始数据集。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>id</th>\n",
       "      <th>MSSubClass</th>\n",
       "      <th>MSZoning</th>\n",
       "      <th>LotFrontage</th>\n",
       "      <th>NumRooms</th>\n",
       "      <th>Alley</th>\n",
       "      <th>Price</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>60.0</td>\n",
       "      <td>RL</td>\n",
       "      <td>65</td>\n",
       "      <td>NaN</td>\n",
       "      <td>Pave</td>\n",
       "      <td>127500</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>70.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>70</td>\n",
       "      <td>2.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>106000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>80</td>\n",
       "      <td>4.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>178100</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>90</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>140000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   id  MSSubClass MSZoning  LotFrontage  NumRooms Alley   Price\n",
       "0   1        60.0       RL           65       NaN  Pave  127500\n",
       "1   2        70.0      NaN           70       2.0   NaN  106000\n",
       "2   3         NaN      NaN           80       4.0   NaN  178100\n",
       "3   4         NaN      NaN           90       NaN   NaN  140000"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import os\n",
    "import pandas as pd\n",
    "\n",
    "os.makedirs(os.path.join('..', 'data'), exist_ok=True)\n",
    "data_file = os.path.join('..', 'data', 'house_tiny.csv')\n",
    "with open(data_file, 'w') as f:\n",
    "    f.write('id,MSSubClass,MSZoning,LotFrontage,NumRooms,Alley,Price\\n')  # 列名\n",
    "    f.write('1,60,RL,65,NA,Pave,127500\\n')  # 每行表示一个数据样本\n",
    "    f.write('2,70,NA,70,2,NA,106000\\n')\n",
    "    f.write('3,NA,NA,80,4,NA,178100\\n')\n",
    "    f.write('4,NA,NA,90,NA,NA,140000\\n')\n",
    "    \n",
    "original_data = pd.read_csv(data_file)\n",
    "original_data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.2.1\n",
    "\n",
    "删除缺失值最多的列。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：** "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;通过观察，可以看到 MSZoning 和 Alley 列的缺失值一样多，需要考虑到缺失值最多的列有多个的情况使用`original_data.isna().sum()[original_data.isna().sum() == original_data.isna().sum().max()].index.tolist()`得到最多缺失值的列名, 再进行删除"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['MSZoning', 'Alley']"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "original_data.isna().sum()[original_data.isna().sum() == original_data.isna().sum().max()].index.tolist()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>id</th>\n",
       "      <th>MSSubClass</th>\n",
       "      <th>LotFrontage</th>\n",
       "      <th>NumRooms</th>\n",
       "      <th>Price</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>60.0</td>\n",
       "      <td>65</td>\n",
       "      <td>NaN</td>\n",
       "      <td>127500</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>70.0</td>\n",
       "      <td>70</td>\n",
       "      <td>2.0</td>\n",
       "      <td>106000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>NaN</td>\n",
       "      <td>80</td>\n",
       "      <td>4.0</td>\n",
       "      <td>178100</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "      <td>NaN</td>\n",
       "      <td>90</td>\n",
       "      <td>NaN</td>\n",
       "      <td>140000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   id  MSSubClass  LotFrontage  NumRooms   Price\n",
       "0   1        60.0           65       NaN  127500\n",
       "1   2        70.0           70       2.0  106000\n",
       "2   3         NaN           80       4.0  178100\n",
       "3   4         NaN           90       NaN  140000"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = original_data.drop(original_data.isna().sum()[original_data.isna().sum() == original_data.isna().sum().max()].index.tolist(), axis=1)\n",
    "data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.2.2\n",
    "\n",
    "将预处理后的数据集转换为张量格式。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：** "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;使用`torch.tensor`方法，将数据集转换为张量格式。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>id</th>\n",
       "      <th>MSSubClass</th>\n",
       "      <th>LotFrontage</th>\n",
       "      <th>NumRooms</th>\n",
       "      <th>Price</th>\n",
       "      <th>Alley_Pave</th>\n",
       "      <th>Alley_nan</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>60.0</td>\n",
       "      <td>65</td>\n",
       "      <td>NaN</td>\n",
       "      <td>127500</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>70.0</td>\n",
       "      <td>70</td>\n",
       "      <td>2.0</td>\n",
       "      <td>106000</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>NaN</td>\n",
       "      <td>80</td>\n",
       "      <td>4.0</td>\n",
       "      <td>178100</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "      <td>NaN</td>\n",
       "      <td>90</td>\n",
       "      <td>NaN</td>\n",
       "      <td>140000</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   id  MSSubClass  LotFrontage  NumRooms   Price  Alley_Pave  Alley_nan\n",
       "0   1        60.0           65       NaN  127500           1          0\n",
       "1   2        70.0           70       2.0  106000           0          1\n",
       "2   3         NaN           80       4.0  178100           0          1\n",
       "3   4         NaN           90       NaN  140000           0          1"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 将Alley列转换成one-hot编码\n",
    "data = pd.get_dummies(data, dummy_na=True)\n",
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[1.0000e+00, 6.0000e+01, 6.5000e+01,        nan, 1.2750e+05, 1.0000e+00,\n",
       "         0.0000e+00],\n",
       "        [2.0000e+00, 7.0000e+01, 7.0000e+01, 2.0000e+00, 1.0600e+05, 0.0000e+00,\n",
       "         1.0000e+00],\n",
       "        [3.0000e+00,        nan, 8.0000e+01, 4.0000e+00, 1.7810e+05, 0.0000e+00,\n",
       "         1.0000e+00],\n",
       "        [4.0000e+00,        nan, 9.0000e+01,        nan, 1.4000e+05, 0.0000e+00,\n",
       "         1.0000e+00]], dtype=torch.float64)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 将数据集转换为张量格式\n",
    "data=torch.tensor(data.values)\n",
    "data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2.3 线性代数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.3.1\n",
    "\n",
    "证明一个矩阵$\\mathbf{A}$的转置的转置是$\\mathbf{A}$，即$(\\mathbf{A}^\\top)^\\top = \\mathbf{A}$。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;假设存在$A^T = B, B^T = C$，根据矩阵转置的定义，可得：\n",
    "$$\n",
    "b_{ij} = a_{ji}，c_{ij} = b_{ji}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "易知：\n",
    "$$\n",
    "c_{ij} = b_{ji} = a_{ij}\n",
    "$$\n",
    "则：\n",
    "$$\n",
    "c_{ij} = a_{ij}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "根据矩阵定义，可得：\n",
    "$$\n",
    "C = A = B^T = (A^T)^T\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;以下使用`torch`编程进行验证："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[True, True, True, True],\n",
       "        [True, True, True, True],\n",
       "        [True, True, True, True],\n",
       "        [True, True, True, True],\n",
       "        [True, True, True, True]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = torch.arange(20).reshape(5, 4)\n",
    "A.T.T==A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.3.2 \n",
    "\n",
    "给出两个矩阵$\\mathbf{A}$和$\\mathbf{B}$，证明“它们转置的和”等于“它们和的转置”，即$\\mathbf{A}^\\top + \\mathbf{B}^\\top = (\\mathbf{A} + \\mathbf{B})^\\top$。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;假设$C = A + B$，根据矩阵运算的定义可得：\n",
    "$$\n",
    "c_{ij} = a_{ij} + b_{ij}\n",
    "$$\n",
    "对矩阵$C$进行转置，将$i,j$进行互换，可得：\n",
    "$$\n",
    "c_{ij} = a_{ji} + b_{ji}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "根据矩阵定义，可得：\n",
    "$$\n",
    "C^T = A^T + B^T = (A + B)^T\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;以下使用`torch`编程进行验证："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[True, True, True, True],\n",
       "        [True, True, True, True],\n",
       "        [True, True, True, True],\n",
       "        [True, True, True, True]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = torch.rand(4,4)\n",
    "B=torch.rand(4,4)\n",
    "A.T+B.T==(A+B).T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.3.3\n",
    "\n",
    "给定任意方阵$\\mathbf{A}$，$\\mathbf{A} + \\mathbf{A}^\\top$总是对称的吗？为什么？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：**  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;假设$B = A + A^T$，根据矩阵运算的定义可得：\n",
    "$$\n",
    "b_{ij} = a_{ij} + a_{ji}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "对矩阵$B$进行转置，将$i,j$进行互换，可得：\n",
    "$$\n",
    "b_{ji} = a_{ji} + a_{ij}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "根据矩阵定义，可得：\n",
    "$$\n",
    "B^T = A^T + A = (A + A^T)^T\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "所以，根据对称矩阵的定义，给定任意方阵$A$，$A+A^T$是对称矩阵。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.3.4\n",
    "\n",
    "本节中定义了形状$(2,3,4)$的张量`X`。`len(X)`的输出结果是什么？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = torch.arange(24).reshape(2, 3, 4)\n",
    "len(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.3.5 \n",
    "\n",
    "对于任意形状的张量`X`,`len(X)`是否总是对应于`X`特定轴的长度?这个轴是什么?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：** "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`len(X)`对应于`X`的轴0的长度，以下使用代码验证："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([ True, False])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = int(torch.rand(1) * 10) + 1\n",
    "m = int(torch.rand(1) * 10) + 1\n",
    "X = torch.rand(n, m)\n",
    "torch.tensor([len(X), len(X)]) == torch.tensor(X.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.3.6\n",
    "\n",
    "运行`A/A.sum(axis=1)`，看看会发生什么。请分析一下原因？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The size of tensor a (4) must match the size of tensor b (5) at non-singleton dimension 1\n"
     ]
    }
   ],
   "source": [
    "A = torch.arange(20).reshape(5, 4) \n",
    "try:\n",
    "    A/A.sum(axis=1)\n",
    "except Exception as e:\n",
    "    print(e)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "代码会报错, `A.sum(axis=1)`得到的结果比`A`少一个轴且与`A`的轴1长度不一致，无法广播，可以试着加上`keepdims=True`以达到想要的结果。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[0.0000, 0.1667, 0.3333, 0.5000],\n",
       "        [0.1818, 0.2273, 0.2727, 0.3182],\n",
       "        [0.2105, 0.2368, 0.2632, 0.2895],\n",
       "        [0.2222, 0.2407, 0.2593, 0.2778],\n",
       "        [0.2286, 0.2429, 0.2571, 0.2714]])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = torch.arange(20).reshape(5, 4) \n",
    "A/A.sum(axis=1,keepdims=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.3.7\n",
    "\n",
    "考虑一个具有形状$(2,3,4)$的张量，在轴0、1、2上的求和输出是什么形状？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "通过代码验证各轴上的求和输出的形状："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "在轴0上的求和：torch.Size([3, 4])\n",
      "在轴1上的求和：torch.Size([2, 4])\n",
      "在轴2上的求和：torch.Size([2, 3])\n"
     ]
    }
   ],
   "source": [
    "A = torch.arange(24).reshape(2,3,4) \n",
    "print(f'在轴0上的求和：{A.sum(axis=0).shape}\\n在轴1上的求和：{A.sum(axis=1).shape}\\n在轴2上的求和：{A.sum(axis=2).shape}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.3.8\n",
    "\n",
    "为`linalg.norm`函数提供3个或更多轴的张量，并观察其输出。对于任意形状的张量这个函数计算得到什么？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：** 在哪个轴上求和哪个轴就消失 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = torch.randn(2, 3, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "torch.Size([3, 4])\n",
      "torch.Size([2, 4])\n",
      "torch.Size([2, 3])\n"
     ]
    }
   ],
   "source": [
    "# 计算不同维度上的范数\n",
    "norm_dim0 = torch.linalg.norm(x, dim=0)\n",
    "print(norm_dim0.shape)\n",
    "\n",
    "norm_dim1 = torch.linalg.norm(x, dim=1)\n",
    "print(norm_dim1.shape)\n",
    "\n",
    "norm_dim2 = torch.linalg.norm(x, dim=2)\n",
    "print(norm_dim2.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2.4 微积分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.4.1 \n",
    "\n",
    "绘制函数$\\displaystyle y = f(x) = x^3 - \\frac{1}{x}$和其在$x = 1$处切线的图像。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 定义函数\n",
    "def f(x):\n",
    "    return x**3 - 1/x\n",
    "\n",
    "# 定义x=point的切线\n",
    "def f_tangent(f, x, point):\n",
    "    h = 1e-4\n",
    "    grad = (f(point+h) - f(point)) / h\n",
    "    return grad*(x-point) + f(point)\n",
    "\n",
    "# 绘制函数图像和切线\n",
    "x = np.arange(0.1, 2.0, 0.01)\n",
    "y = f(x)\n",
    "y_tangent = f_tangent(f, x=x, point=1)\n",
    "plt.plot(x,y, label='f(x)')\n",
    "plt.plot(x,y_tangent, label='Tangent line at x=1')\n",
    "plt.legend()\n",
    "plt.title('Graph of f(x) and its tangent line at x=1')\n",
    "plt.xlabel(\"X\")\n",
    "plt.ylabel(\"Y\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.4.2\n",
    "\n",
    "求函数$f(\\mathbf{x}) = 3x_1^2 + 5e^{x_2}$的梯度。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;根据梯度的定义：\n",
    "$$\n",
    "\\frac{\\partial{f(x)}}{\\partial{x_1}} = 6 x_1 \\\\\n",
    "\\frac{\\partial{f(x)}}{\\partial{x_2}} = 5 e^{x_2}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "则：\n",
    "$$\n",
    "\\frac{\\partial{f(x)}}{\\partial x} = (6 x_1, 5 e^{x_2})\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;利用代码编程验证结果："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f(x)的梯度： tensor([ 0.0000, 13.5914])\n",
      "验证梯度： tensor([True, True])\n"
     ]
    }
   ],
   "source": [
    "x = torch.arange(2.0, requires_grad=True)\n",
    "y = 3*x[0]**2+ 5*torch.exp(x[1])\n",
    "y.backward()\n",
    "\n",
    "print(\"f(x)的梯度：\", x.grad)\n",
    "print(\"验证梯度：\", x.grad==torch.tensor([6*x[0], 5*torch.exp(x[1])]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.4.3 \n",
    "\n",
    "函数$f(\\mathbf{x}) = \\|\\mathbf{x}\\|_2$ 的梯度是什么？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;根据2-范数的定义，可得：\n",
    "$$\n",
    "f(x) = \\|x\\|_2 = \\sqrt{x_1^2 + x_2^2 + \\cdots + x_n^2}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "函数$f(x)$的梯度：\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\frac{\\partial{f}}{\\partial{x}} \n",
    "&= \\left( \\frac{\\partial{f}}{\\partial{x_1}}, \\frac{\\partial{f}}{\\partial{x_2}}, \\cdots, \\frac{\\partial{f}}{\\partial{x_n}}  \\right) \\\\\n",
    "&= \\left(\\frac{1}{2} \\cdot 2 x_1 \\cdot \\left( \\sum_{i=1}^n x_i^2 \\right)^{-\\frac{1}{2}}, \\frac{1}{2} \\cdot 2 x_2 \\cdot \\left( \\sum_{i=1}^n x_i^2 \\right)^{-\\frac{1}{2}}, \\cdots, \\frac{1}{2} \\cdot 2 x_n \\cdot \\left( \\sum_{i=1}^n x_i^2 \\right)^{-\\frac{1}{2}} \\right) \\\\\n",
    "&= \\left( \\frac{x_1}{\\sqrt{\\sum_{i=1}^n x_i^2}},  \\frac{x_2}{\\sqrt{\\sum_{i=1}^n x_i^2}}, \\cdots, \\frac{x_n}{\\sqrt{\\sum_{i=1}^n x_i^2}} \\right) \\\\\n",
    "&= \\left( \\frac{x_1}{\\|x\\|_2}, \\frac{x_2}{\\|x\\|_2}, \\cdots, \\frac{x_n}{\\|x\\|_2} \\right) \\\\\n",
    "&= \\frac{x}{\\|x\\|_2}\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "利用代码编程验证结果："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f(x)的梯度： tensor([0.0000, 0.2673, 0.5345, 0.8018])\n",
      "验证结果： tensor([True, True, True, True])\n"
     ]
    }
   ],
   "source": [
    "x = torch.arange(4.0,requires_grad=True)\n",
    "y = torch.dot(x, x).sqrt()\n",
    "y.backward()\n",
    "print(\"f(x)的梯度：\", x.grad)\n",
    "print(\"验证结果：\", x.grad==x/(torch.dot(x, x).sqrt()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.4.4\n",
    "\n",
    "尝试写出函数$u = f(x, y, z)$，其中$x = x(a, b)$，$y = y(a, b)$，$z = z(a, b)$的链式法则。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：** "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;根据链式求导法则，可得：\n",
    "$$\n",
    "\\frac{\\partial u}{\\partial a}=\\frac{\\partial u}{\\partial x} \\frac{\\partial x}{\\partial a}+\\frac{\\partial u}{\\partial y} \\frac{\\partial y}{\\partial a}+\\frac{\\partial u}{\\partial z} \\frac{\\partial z}{\\partial a} \\\\\n",
    "\\frac{\\partial u}{\\partial b}=\\frac{\\partial u}{\\partial x} \\frac{\\partial x}{\\partial b}+\\frac{\\partial u}{\\partial y} \\frac{\\partial y}{\\partial b}+\\frac{\\partial u}{\\partial z} \\frac{\\partial z}{\\partial b}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2.5 自动微分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.5.1 \n",
    "\n",
    "为什么计算二阶导数比一阶导数的开销要更大？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：** "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;在 PyTorch 中计算一阶导数和二阶导数都是通过自动微分机制实现的，但是计算二阶导数的开销通常要比计算一阶导数更大，原因如下：\n",
    "\n",
    "1. 计算图的构建：在计算一阶导数时，PyTorch 只需要构建一次计算图，然后根据需要进行反向传播即可。而在计算二阶导数时，需要构建两次计算图，第一次是计算一阶导数，第二次是计算一阶导数的导数，也就是二阶导数。因此，计算二阶导数需要更多的计算图构建操作。\n",
    "2. 内存占用：计算一阶导数时，PyTorch 只需要保留一阶导数的梯度值，而计算二阶导数时需要保留一阶导数和二阶导数的梯度值，这会占用更多的内存。\n",
    "3. 计算量增加：计算二阶导数需要对一阶导数进行额外的计算，这增加了计算量和计算时间。\n",
    "\n",
    "因此，计算二阶导数的开销通常要比计算一阶导数更大。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.5.2 \n",
    "\n",
    "在运行反向传播函数之后，立即再次运行它，看看会发生什么。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([0., 1., 2., 3.], requires_grad=True)\n",
      "tensor([0.0000, 0.2673, 0.5345, 0.8018])\n",
      "Trying to backward through the graph a second time (or directly access saved tensors after they have already been freed). Saved intermediate values of the graph are freed when you call .backward() or autograd.grad(). Specify retain_graph=True if you need to backward through the graph a second time or if you need to access saved tensors after calling backward.\n"
     ]
    }
   ],
   "source": [
    "x = torch.arange(4.0,requires_grad=True)\n",
    "y = torch.dot(x, x).sqrt()\n",
    "print(x)\n",
    "y.backward()\n",
    "print(x.grad)\n",
    "try:\n",
    "    y.backward()\n",
    "    print(x.grad)\n",
    "except Exception as e:\n",
    "    print(e)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "根据上述运行结果，在运行反向传播函数之后，立即再次运行它，会报错，提示“尝试第二次反向传播（或者在释放保存的张量后直接访问它们）。调用.backward()或autograd.grad()时，将释放图中保存的中间值。如果需要第二次执行反向传播，或者在反向传播调用后需要访问保存的张量，请指定retain_graph=True。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([0., 1., 2., 3.], requires_grad=True)\n",
      "tensor([0.0000, 0.2673, 0.5345, 0.8018])\n",
      "tensor([0.0000, 0.5345, 1.0690, 1.6036])\n"
     ]
    }
   ],
   "source": [
    "x = torch.arange(4.0, requires_grad=True)\n",
    "y = torch.dot(x, x).sqrt()\n",
    "print(x)\n",
    "y.backward(retain_graph=True)\n",
    "print(x.grad)\n",
    "try:\n",
    "    y.backward()\n",
    "    print(x.grad)\n",
    "except Exception as e:\n",
    "    print(e)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.5.3 \n",
    "\n",
    "在控制流的例子中，我们计算`d`关于`a`的导数，如果将变量`a`更改为随机向量或矩阵，会发生什么？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：** "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "grad can be implicitly created only for scalar outputs\n"
     ]
    }
   ],
   "source": [
    "def f(a):\n",
    "    b = a * 2\n",
    "    while b.norm() < 1000:\n",
    "        b = b * 2\n",
    "    if b.sum() > 0:\n",
    "        c = b\n",
    "    else:\n",
    "        c = 100 * b\n",
    "    return c\n",
    "a = torch.randn(size=(1, 3), requires_grad=True)\n",
    "d = f(a)\n",
    "try:\n",
    "    d.backward()\n",
    "except Exception as e:\n",
    "    print(e)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "根据上述执行结果，如果将变量`a`更改为随机向量或矩阵，会报错，因为PyTorch只能对标量结果求梯度，可以将代码进行以下修改："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[1024., 1024., 1024.]])\n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    d.backward(torch.ones_like(d))\n",
    "except Exception as e:\n",
    "    print(e)\n",
    "print(a.grad)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.5.4 \n",
    "\n",
    "重新设计一个求控制流梯度的例子，运行并分析结果。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：** "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor(-1.3132, requires_grad=True)\n",
      "tensor(102400.)\n"
     ]
    }
   ],
   "source": [
    "def f(a):\n",
    "    b = a * 2\n",
    "    while b.norm(1) < 1000:\n",
    "        b = b * 2\n",
    "    if b.sum() > 0:\n",
    "        c = b\n",
    "    else:\n",
    "        c = 100 * b\n",
    "    return c.sum()\n",
    "a = torch.randn(size=(), requires_grad=True)\n",
    "d = f(a)\n",
    "print(a)\n",
    "d.backward()\n",
    "print(a.grad)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.5.5\n",
    "\n",
    "使$f(x)=\\sin(x)$，绘制$f(x)$和$\\displaystyle \\frac{df(x)}{dx}$的图像，其中后者不使用$f'(x)=\\cos(x)$。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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dgG3Pi+35vxRyGwbDeBYrzo01Bi66V2yv/a2o1/ADByoaWLm7HJMJ7g/gRRrg/10+krAQM1sKa/ns0MmAvrbCQDRWwlbPRfrS5YFtUc77npiUXnMY9r7lt5fR6gUXTU5nVEqM317ndIYnRnH9jEwA1XCh6J21vwGnHXIuCWqV7t5QzpKszLhVqBM3VUC+f2ZW/WWt0FS6amLaeY1SGAhptgi+PXsYAH9YfUhFlxQ9s+EpcLSKDrURlwf2ta0xMPsuse2n6NKeknpW76/EbIL/d/konx//XNx5yUgsZhPrD1ezs+hUwF9fYQBOnYDdb4jty/WdMDEQlLMkKyHWztqlL57y+dy4EzXNvL+rDIA7Lhnh02P3le9dPAJriJldJfVsPq7SAIrT6BpVuuR+fYTvZt3eJbrk+9qlP3oiOounZDDCRyKw/SErIZKvTs0ARKOHQnEGG/8slLpHXCaaHwyKcpZkZurNYlZV3Qmfd8b9dd0xXG64eHQSEzP06WoYEm31pgH+6lEOVyi8bH5GRJUyZgQ+qqQRHtspfvn5kz6V8zh6somPCyoxmeD7l4302XH7y/cvHYnZBJ8cqGJvab1udiiCkKaTsOMlsT3vR/raMkCUsyQzYZGibgJ8eqGuamjjX9tKAHGh1JPb5uVgNsGnB09yoKJBV1sUQYS9Bbb/XWzP/aG+4xRm3ibkPKr2CfViH/HC58cBuHxsii5RJY3sxCiumZwOCP0zhcLLlr+Cow3Sp0H2hXpbMyCUsyQ72oW6cg8cXeOTQ/5j0wnsThczhsUza3iCT455vmQnRnHlRDFa5W/rjulqiyKI2P06tJ4SI4DGXq2vLRHxMPUmsb3p/3xyyNpmO//aLhYsyy70k8BmP1g6T9iwcncZVQ1tOlujCAram2DL38T2vB8ZYv5bbyhnSXYiE2DaErG9+W8DPly7w8lrnplQ35mn/0Ua4PaLcgB4L19dqBUIEchNfxHbed/zz1iT/pL3PcAEhz+CkwNXvX550wnaHS5yM2y6L1gAJmXGMX1YPB1ONy9vOqG3OYpgYM+b0FYP8cP1X7D4AOUsDQZm3ibuD38kpq4PgP/sKae6yU6aLZwrxqcM3DYfMDlLXKgdLjevby3W2xyF3hxdA9WHICxG1O0FA0NGwJgrxfbmvwzoUHaHi5c2CofktguHYwqSFft35orF0yubi2jr8J+ulMIAuN2w5TmxPfO24FiwDBDlLA0GEkdCzqWAG7a9MKBDvbhBXKRvyhtKiCV4Pj7fukDICLy2pQiH0//jJRRBzOa/ivtpS0SBdbBwwZ3iPv81seI+Tz7aX0F1UzvJMVauyk3zkXEDZ8GEFNJt4dQ023nP0ymrGKQUbRI1eiERnSlogxM833YK/zLrdnG/4x9i8vN5kF9cx67iOsIsZm6cNdSHxg2cK3NTSYgKo7y+jTUHqvQ2R6EXdUVw5GOxPXOpvracTvY8SBorOvT2/PO8D/PqZpEGv2FmFqFBtGAJsZj51uxsQESXFIOYrc+K+9yviZo9CQieM03hX0YvECKVrbWw7/zUhLVahC9PSiMx2upL6waMNcTiHYGiaiYGMTtfBtww/CKR+gomTCaYfovY3v7ieXWnHq9uZsPRGkwm4SwFG9fPyCTEbGJXcR0F5ao7dVDSWAn73xPbs5bpa4sPUc7SYMFsEareIC7U/aSp3cEHu8sB+EZecEWVNG7KG4rJBOsPV1NY3ay3OYpA43SIyCl0OiXBxqQbwGKFij1QtrPfT9eaKy4ZnURmfAAGAveTxGgrX/LUMr6h6gcHJ/mvgKtDqOanTdbbGp+hnKXBxJRvgskCxZuhun9qux/sLqO1w0lOYhQzhgVnWDUrIZKLRycBeNuqFYOII6uhsQwiEmDsl/W2pmciE2D8tWK7n4uWdofT+7n+Zt4wHxvmO7QU/ds7S1Wh92DD7Yb8V8W21oUtCcpZGkzEpMLI+WK7n/Pi3vSIUF4/Iytoum964vrpIjXx7x0lOF1qXtygYvsKcT/lm2LcT7Ay/dvifu+/ob2xz0/7eH8Vtc12UmPDuXRMkp+MGzjzRiaSERdBfWsHH+6r0NscRSAp2SpG+4RGwoSv6G2NT1HO0mBjyjfF/a7X+zzY80hVE9tPnMJiNnHdtAw/GjdwLh+XTGx4COX1bWw8WqO3OYpA0VQlpDEApn1bX1vOxbC5MGQk2Js6azv6wFs7xILluukZQdWJejoWs8k7huj1LSoVN6jY+bK4H7dIDJKWiOA94xT+YcyVojuhsQyOftqnp/xzu7jgXTI6ieTYcH9aN2DCQy0smiJGL/xru7pQDxr2/lsM68yYDkmj9bamd0wmmHSj2NamsZ+D6qZ21h46CcBXpmb6yzKfISLQsPFYDaV1rXqbowgE9hbY62kekkQuoCvKWRpshFgh9+tiO//lc+7udLl5e0cpgHe1GOx8zZOKW7Wvgsa2Dp2tUQSEXa+Le80JCXYmec7B4+ugvvScu7+/qwyny83kTBsjk/WbA9dXMuIimJUtlMXfy1eaS4OCgvfB3ghxQ2HYPL2t8TnKWRqMaF7/gf9AW+/tvZuP1VDV2I4tIpTLxgaHYve5mJxpY0RSFG0dLm8Hn0JiTh6E8nwwh8DEr+ptTd+IHwZD5wDuPmkuveVZsHx1mjEWLABfmSpS9u/sPLczqJAALUo6+Ztgls+1kO8dKc5N6iRIHA3OdjjwQa+7akq8V+WmEhZijI+LyWTiuuniS0UpCQ8CtIv0yPkQlaivLf1Biy7tfqNXzaXDlY3sKa0nxGzimsnpATJu4FyZm0aYxczBykaluSQ7zdVwbK3Y1j7XkmGMbz+FbzGZYOLXxPbef511t3aHk//sEZGZRZODu7D7dK6ZJL5UNh6rUcN1Zcblgt2eyMykG/S1pb9MWAyWMKjaD5V7z7rb257IzCVjkkmICguQcQNHRKOTARVdkp59b4uawbQpwScG6yOUszRYyfU4S0c/FauCHvjs4Eka2hykxoYHxWTz/pCVEMnUoXG43XgdPoWElGyB+iKwxnYOqjUKEfEweqHY3v1mj7u43W5WelLJWuOCkVg8Vdj8bn6ZkvKQmb3/Fvfa94qEKGdpsDJkhFgFuJ2w/50ed3nXk8K6ZnIaFnPwaiudDS269L6qW5KXfW+L+7FXQ2iEvracDxOvE/f73+kxFbe3tIGi2hbCQ81c7onSGIlLxyYTEx5CRUMb2wpr9TZH4Q/qiqFoI2CCCQapGTwPlLM0mNFWAXvPnBXX3O5gTUElYLwUnMbVk9IwmWD7iVOUnGrR2xyFr3G5YP+7Ynv8Yl1NOW9GXSEE/OqKehx/snKPWLBcPjaFKGtIoK0bMNYQC1eMTwVUhFdatAXLsDlgM+Z3RV8wnLP09NNPk52dTXh4OHl5eWzZsuWs+15yySWYTKYzbldffbV3n1tuueWM3y9cuDAQb0V/JnwVMMGJL85oX/7kQBVtHS6yh0QyMSNWH/sGSEpsOHme9KHqipOQ4s3QWA5WG4y4VG9rzo+wSDHkGjq/dDy43W7v5/bqSWmBtsxnXJUrnKX/7q3ApVJx8qGl4LQoqaQYyll64403uOeee3jooYfYsWMHkydPZsGCBVRVVfW4/1tvvUV5ebn3tnfvXiwWC9dff323/RYuXNhtv9deey0Qb0d/bBkw9AKxfWBlt1+t2ivGFFyZmxbU403OhdY9tFI5S/KhpY/HXhXc403OhRYVOy0Vt6uknpJTrUSEWrh0jPFScBrzRiUSYw2hqrGd7UWn9DZH4UtOFQrZDpO5c+ahpBjKWXriiSdYtmwZt956K+PHj+eZZ54hMjKSF154ocf9ExISSE1N9d5Wr15NZGTkGc6S1Wrttl98fHAOivUL464R9wXvex9q63Dy6UHhgF45MVUPq3zGwgmpmE2wp7RepeJkQoYUnEa3VNwO78Mf7Pak4MYlExFm0cu6AWMNsTB/vNBoU6k4ySjwLLKHzTWWbMd5YBhnyW63s337dubPn+99zGw2M3/+fDZu3NinYzz//PPceOONREVFdXt87dq1JCcnM2bMGO644w5qanqfKdbe3k5DQ0O3m2HRprOf+MLbFffZoZO02J1kxEWQm2HT0biBMyTaygyPkvBH+yp1tkbhM2RIwWl0S8W9A4gU3H/2iOju1bnGTcFpXOV5D//do1JxUqEtssct0teOAGAYZ6m6uhqn00lKSncV6ZSUFCoqzj3ZesuWLezdu5fbbrut2+MLFy7kpZdeYs2aNfz2t7/ls88+48orr8TpPPuQ2cceewybzea9ZWVlnd+bCgbih3m64lxegUotBbdwYqqhU3AaCyeI6NgqNQFdHrSL9JgrjZ2C09CiYwXvg9vN/vIGSutaCQ81c4mBU3AaF45KJNoquuJ2FqtUnBQ0VohFC4huVMkxjLM0UJ5//nlyc3OZNWtWt8dvvPFGFi1aRG5uLosXL2blypVs3bqVtWvXnvVYy5cvp76+3nsrLjb4wNYuqbh2h5OPPV1wRk/BaVwxQTjYWwtrqW5q19kaxYBxuztr7LTPrtEZOR8sVjh1HKoKvFHQi0YlGToFpxEeauHyccLpUxFeSTiwEnBDxgypu+A0DOMsJSYmYrFYqKzsfqJVVlaSmtr7l3pzczOvv/46S5cuPefr5OTkkJiYyJEjR866j9VqJTY2ttvN0Ggh1GNr2VJwnMY2B8kxVqYNlaN2KzM+ktwMG243fLxfXagNT+U+qDsBIeHGT8FpWKMh5xKxffADPvJ8Tq+YIMeCBeBLnrql1eoclIP974n78fKn4MBAzlJYWBjTp09nzZo13sdcLhdr1qxh9uzZvT73n//8J+3t7dx8883nfJ2SkhJqampISzN+nUCfSRoNiWPA1UHZVlE0+6XxKZgNKER5NhZ4oksqFScBB/8j7kdcBmFRve9rJDypDPve9ykob8BswpBClGfj4tFJhFpMHKtu5khVk97mKAZCSy0Ufi62ZYnungPDOEsA99xzD88++ywrVqygoKCAO+64g+bmZm699VYAlixZwvLly8943vPPP8/ixYsZMmRIt8ebmpq499572bRpE4WFhaxZs4Zrr72WkSNHsmDBgoC8p6BhnCj0Tij5GOhcBcrCQk9K8Ysj1TS2dehsjWJAaCm4MVfpa4evGXMlYCKsahep1DBreALxBpoFdy5iwkOZPUJ0TKnoksE5vFpMf0ieAAk5elsTEAzlLN1www08/vjjPPjgg0yZMoX8/HxWrVrlLfouKiqivLx7a+rBgwf5/PPPe0zBWSwWdu/ezaJFixg9ejRLly5l+vTprF+/HqtVgqLR/jBazNXKc+7EFubmgpwh53iCsRiZHMPwxCg6nG7WH+55Fp7CANQVQ/kuoetitFlw5yI6GbJETeV8yw4WSJSC0+hMxakIr6E59F9xP2aQCDgDhtPPv+uuu7jrrrt6/F1PRdljxozB3cPMJYCIiAg+/PBDX5pnXDKm0xyaQGxHLbeklxMeavyi0tO5fGwyz31+nDUFVd5WZoXBOOi5SGflSanr0pKzkMjizVxh3kaOZNFdgC+NS+EX7+xlZ3EdJxvbSYoZZItSGXDY4YinHGa0ZAuWXjBUZEnhR8xmvjBNA+DL4fn62uInLh8nvnw+PVilJqAblYNC3kK6FJyHL0JEZGmOZT+ZEQ6drfE9qbZwJmWKZgtt9qTCYBRtgPYGiEqCjOl6WxMwlLOkAKCsrpV/N+UCkFO7vscJ6EZnRnY8MeEh1DbbyS+u09scRX9pb4TCL8S2bCk4D++VRHHUlUYITjj2qd7m+IUveRYtHytnyZgcXCXuRy0A8+BxIQbPO1X0ypoDVax35dJBCJb6E3DyoN4m+ZxQi5mLRycBalVrSI59Bq4OiB8OQ0bqbY3PcThdfHawik9dU8QDhz7S1R5/cZlHb+mLIzW0dZxd/FcRhLjdg7JeCZSzpPCwpqCSFsIpT/CIdmonhGTM96xqPznQ8/BlRRBz2FNfOHoBSKAsfzrbT5yioc3BttAZ4oEjq8UMPMkYnxZLSqyV1g4nW47X6m2Ooj+cPCiG51rCIEcSjbM+opwlBW0dTjYeFfPwrOM9svWH5Cx8v3h0EmYTHKhoVIN1jYTbLdqVAUZ9SV9b/MQnnuHV0WMuhrBoaKqEil06W+V7TCYTl3pGuGgDuxUGQVuwZF8ohFQHEcpZUrDpWA3tDhdptnCSp3sG6xZvgdY6Xe3yB/FRYcwYJgbrfqqiS8ahYo8YnBsaCcPm6W2NX/ikQHweLxqX0anmLWkqTpt3t/bgSZ0tUfQL74LlCn3t0AHlLCm8F6xLxiRhis+GxNFCcOzYWl3t8hcXjxF1S58dUnpLhkFb0eZcAqHhupriD4prWzhc1YTFbOLiUUmdX0aH5Yzwzh05hFCLiePVzRyvbtbbHEVfaG+Eok1iW9Lobm8oZ0nBZ4eEs3TxaM9ohZHzxf2R1TpZ5F+0Iu8NR6tpd6gCU0OgRVgkXdGu9ZyD04fGY4sM7XyfpTugSb7oS0x4KDOzRYR3rUrFGYPj67o0WIzQ25qAo5ylQc6JGrGyCzGbmDvSo9rtdZbWSCkhMD4tlsRoKy12J9sLT+ltjuJctNRC6TaxLemK9vPDwiG6aLRHaDM2DVJzATcc/UQ/w/xIZ92SfM6glGgpOO37YZChnKVBjpaCm5mdQEx4qHhw2FwIiRA1IpX7dLTOP5jNJm90SYuqKYKY45+B2wVJ48CWqbc1PsfhdLHhiGiwuHBUUucvRlwm7iXVW7p0rHivm44pCYGgx+3uVO2WdMFyLpSzNMjRQuCXjOlykQ4Nh+EXiu0jH+tglf/prFtSzlLQo12kNedBMnaV1NPY7sAWEcrEDFvnL7TW7KOfShnhHZEUTZotHLvDxdZCJSEQ1FQfgvoisFghW84Gi3OhnKVBTLvDycZjYkV7cVdnCbqk4uR0li4cmYjJIyFQUd+mtzmKs+HukoYaKaez9LlnsPPckUOwmLvoRw2dDSHh0FQBJw/oZJ3/MJlMzB0p0o6fq+HWwY2Wghs2B8Ki9LVFJ5SzNIjZXniKtg4XKbFWxqTEdP+l5iwVbYT2psAb52fio8KYnBkHwGeHVIFp0FJ9CBpKhdMwbK7e1viF9Z56pW4pOBAR3qGzxfZROVNxF47yOEtHlLMU1HgXLIOzXgmUszSoWedd0SZiOl0ROSEHbEPB5RAOk4RodUvrlIRA8KKl4IbNgdAIfW3xA41tHez0zCmc54mydGOEJxUnad3SnBHiPe8ra6CmqV1naxQ94miHExvE9ojBpdrdFeUsDWI+P+LpwDl9RQtinETOxWJbUr2leZ5V7Yaj1bhc8tWESMFRueuVNh2rxelyMzwxiqyEyDN30OqWCr8Ahz2wxgWApBgrY1NFVPsLzxQBRZBRshUcrRCVBMnj9bZGN5SzNEipaWpnb2kDgLdu4Aw0FeFjnwXGqAAzOTOOyDALp1o6KKho0Nscxel0tAknAWDE5fra4ie0FFyPUSWAlIkQmQgdzVCyJYCWBQ4tFfeFqlsKTrTF8vCLpZzJ2FeUszRI0VZx49JiSYqx9rzT8IvEfeUeaJbvQhYWYmbWcCGMp7VuK4KI4k1iRRuTBsnj9LbGL2iFzVqU8wzM5s5Fi6R1S94i7yPVuCXs+jM82mJZyzQMUpSzNEhZf0grKj3LRRogOhmSJ4jt4+sCYFXgmeupmfjiqHzOoOHRPnM5l0i5oi051cKx6mYsZhOzRww5+46S1y3NGp5AmMVMaV0rhTVquHVQ0dYApdvFtua0D1KUszQIcbvd3u6Ts4b/NSSvW5rjUS3fcrwWu8OlszWKbmjOUvaF+trhJ7So0pSsOGI1Qdie0OqWynZCq3yK85FhIUwbFgd0KpkrgoQTX4g5ofHDIW6o3tboinKWBiFHTzZTXt/WLQ11VrTVxHE565bGpcaSEBVGi93JrpI6vc1RaLQ1iLlo0CmQKhnr+7pgsWV4hlu7pI3warIJ61XdUnDhTcFdoqsZwYBylgYhGz0ppxnD4gkPtfS+87A5YA6BU4XiJhlms4nZOSK69IXSegkeijZKvaJ1utzez5t3HlxvaN2AktctbTxWg8OpIrxBw3FVr6ShnKVBiKbarTkJvWKNgYzpYlvSrjgtFaeKvIMILYKiNRlIxr6yeupaOoixhnjFUXslR+66pdwMG7HhITS2OdhdWq+3OQqAxkqo2i+2s+U8D/uDcpYGGS6Xm03HxBymXotKuyJ5Kk4r8t5ZfIoWu0NnaxRA52dNUmdJSzfNHjGEEEsfLsPZczsjvLXH/WucDljMJq9ApZIQCBK0BUvqJIjq43eFxChnaZBxqKqR2mY7EaEWJvVlRQtCXwNEZMklX4h82JBIMuIi6HC62XJcDfTUnZZaqNgjtqV1lvrQjdoVawxkzhLbkjZbaPIJ61U6PDjQPmcqBQcoZ2nQsdGjrzQjO56wkD7++zNnQmgktFR3hmUlwmQyMccTZdugVIT1p3C9uE8aJ+QrJKOtw8mOojqgF0HYnpBcQkArdN9ZdIrmdhXh1RW3u0u90iW6mhIsKGdpkLHJU690QV/qlTRCwkShN8ibivNcqFWRdxAgeb3SzqI67A4XyTFWhif2Y4K7Nx2+XtoIb2a8ivAGBbXHoL4YzKGdw5wHOcpZGkS4XG42H+9nvZLGcMn1ljx/j/3lDZxqlm8Gl6GQ3FnqumA5Y4B1b6RPFRHe1lo4ecBP1umHyWTyRpc2KJFYfdGu81mzIKwfDr3EKGdpEFFQ0UBdSwdRYRZyM2z9e7K2qj2xAZwdPrdNb5JjwxmVHI3b3dktqNCBhnKoPgSYRFGzhGw+fh7RXQBLKGTlie3Cz31sVXCg/U02q8iSvqgU3BkoZ2kQodUrzRyeQGhfOnC6kjIRIoeAvalT/l4yVCouCNDqldImQ0S8vrb4ga71Snk55xCE7YnseeL+hJzOkvY32VtaT2ObfIsyQ+DqIn46XBV3ayhnaRCxqT/6SqdjNnemRWTVW1JF3vojuWRAfrGoV0qKsZLTn3olDc1ZKvxCFOFKRpotgqEJkbjcsO2EfKNdDEHlHjFWJywaMqbpbU3QoJylQYJzIPVKGt5V7Rc+siq4yMsZgtkEx6ubKa9v1ducwYnkK9rNHo2zftcraaRPg5AI0Zl68qCPrQsO8jwjmDapdLg+nNgg7ofOFqlfBWBAZ+npp58mOzub8PBw8vLy2LJly1n3ffHFFzGZTN1u4eHh3fZxu908+OCDpKWlERERwfz58zl8+LC/30bA2VdWT2Obg5jwECak97NeSWOYp4akeAs45CuCtkWEMtFTy6W6cXTgVCHUFQnxxaEX6G2NX9AcgLxzzWQ8GyFhougWJE7FeeqWjqlzUBe0xbDWAa0ADOYsvfHGG9xzzz089NBD7Nixg8mTJ7NgwQKqqqrO+pzY2FjKy8u9txMnTnT7/e9+9zueeuopnnnmGTZv3kxUVBQLFiygra3N328noGj1SnnDE7CYz2NFC5A0VtQtOVqhPN93xgURs7LFl5gqMNUBLaqUMQOs0fra4gdEvZJILfW7uLsr3lScpM6Sx5HcU1qv9JYCjdvdGVnSPmcKwGDO0hNPPMGyZcu49dZbGT9+PM888wyRkZG88MILZ32OyWQiNTXVe0tJSfH+zu128+STT/LAAw9w7bXXMmnSJF566SXKysp45513AvCOAsfG89FXOh2TqVNzQ9JU3CzPhVpFlnTAm4K7UF87/MSu4jraHS4So62MSBpAO7YW4ZW0bikrQSjqO11utqu6pcBy8iC01IhUb9oUva0JKgzjLNntdrZv3878+fO9j5nNZubPn8/GjRvP+rympiaGDRtGVlYW1157Lfv27fP+7vjx41RUVHQ7ps1mIy8vr9djtre309DQ0O0WzHQ4XWw93lkrMSC0C7W2+pCMmZ7I0pGqJqqb2nW2ZhDhdosvf4BsOZ0lLVqZl5NwfvVKGhnTISQcmqugWr6SAeiMLmkyC4oAoS2Cs2aKlK/Ci2GcperqapxOZ7fIEEBKSgoVFRU9PmfMmDG88MILvPvuu7z88su4XC7mzJlDSUkJgPd5/TkmwGOPPYbNZvPesrKyBvLW/M6e0nqa7U5sEaGMT4sd2MG0PHbRJnA5B25ckBEfFcaYlBgAthWq6FLAqDsBjWVCMThzpt7W+IXzUs/vidDwzr+RtHVLHmdJ1S0FFm0RPExOjbOBYBhn6XyYPXs2S5YsYcqUKVx88cW89dZbJCUl8de//nVAx12+fDn19fXeW3FxsY8s9g9di0rN51uvpJGaC9ZYaG/oHHYqGbOGq7qlgKNdpNOnQFikrqb4g3aH05tSmn0++kqn0zUVJyF5w4VDuaukjla7fIuyoMTt7lLcrZyl0zGMs5SYmIjFYqGysrLb45WVlaSmpvbpGKGhoUydOpUjR44AeJ/X32NarVZiY2O73YKZTccGKBnQFbOls1NJ0lScqlvSga7tyhKyu6TeU68UxogkHxSvdy3ylrBuadiQSFJjw+lwutlZpOqWAsKp49BY7onuztDbmqDDMM5SWFgY06dPZ82aNd7HXC4Xa9asYfbsvl1gnU4ne/bsIS0tDYDhw4eTmpra7ZgNDQ1s3ry5z8cMdhxOF9s96SRttTZgtFSc5EXe+8sbaFAqwoGhyFMjKOmKdpO3G/U89ZVOJ3MGWMKgqUIMPZUMk8nkTcVtUouWwKAtWDKmQ2iEvrYEIYZxlgDuuecenn32WVasWEFBQQF33HEHzc3N3HrrrQAsWbKE5cuXe/d/+OGH+eijjzh27Bg7duzg5ptv5sSJE9x2222AOCHvvvtu/ud//of33nuPPXv2sGTJEtLT01m8eLEeb9HnHKhopNnuJMYawpjUGN8ctGuRt4Sr2pTYcLKHROJ2w/ZCtar1O01VUHMEMMHQPL2t8QubvPPgfJCCA/FlptUtaSNiJENb3G1W4pSBwVuvpPSVeiJEbwP6ww033MDJkyd58MEHqaioYMqUKaxatcpboF1UVITZ3On/nTp1imXLllFRUUF8fDzTp09nw4YNjB8/3rvPT3/6U5qbm7n99tupq6tj3rx5rFq16gzxSqOy1RNVmp4df/76SqeTNqX79PPkcb45bhAxa3gChTUtbD5ey6Vjk/U2R260i3TyeCnnwdkdLm+9Ut5Ai7u7MmyuiO4WfgHTb/HdcYMELbK0s7iOtg4n4aEWnS2SHE23S9IB1gPFUM4SwF133cVdd93V4+/Wrl3b7ec//OEP/OEPf+j1eCaTiYcffpiHH37YVyYGFZqzpLXE+4SQMLGqPf6ZuFhL6SwN4c1tJWxRrcv+x5uCkyP1fTq7S+po63CREBXGqGQfim1mz4V1dNYt+SK9F0TkJEaRGG2luqmdXcV1vnU0Fd2pLxEdqSYzZMkZ3R0ohkrDKfqH2+1mqyeN5FNnCaTXW9J0XnaX1KtuHH8jeXG3V19p+AD1lU4nc5Yoxm0sE8W5ktG1bkl1pvqZE54FS9pksPqoXEMylLMkMSdqWjjZ2E6YxcykzPOcB3c2vEXectYtZcZHkGYLx+FS3Th+pa0eKveKbUlrJTS9rhm+XrCERYpiXJBWQuACNVQ3MCjJgHOinCWJ0VJwkzJtvs/3a904jeXSduMovaUAULwF3C6Iz4bYdL2t8TmuLiM7Zmb7oR5Lk/Eo3uT7YwcBWuptR9Ep7A6XztZIjHKWzolyliRmq79WtCC6cbRVraSpOKW3FAC8KTg5o0qHq5poaHMQGWYZuHp+T2ipy6LNvj92EDAyKZq4yFDaOlzsK6vX2xw5aToJ1YfEtuZ8K85AOUsSs81TrzRruJ86jLqm4iREq1tSq1o/Inlxt7ZgmTo0jhCLHy63WbPEfc1haK72/fF1xmw2MX2ouH6pobp+okjrRp0AkX5YWEuCcpYk5WRjO8eqmzGZYPpQP50AkotTjkiKJiEqjHaHiz2ldXqbIx8dbVC6XWxLGv7X6pWmD/PTORiZAEljxXaxnNElLTK+TWme+Qelr9QnlLMkKdtPiIv0mJQYbJGh/nmRrDzRalp3AupL/fMaOmIymZiVreqW/EbZDnDaISoZEnL0tsYvbPNnvZKG1updJGfd0gzP327biVO4JWwm0R3lLPUJ5SxJypbj4iI9w58XaWsMpEwU25KuarW6pa3KWfI93qLS2dJpBAGU17dScqoVswmmDvXjeeitW5LTWcrNsBFmMVPd1M6Jmha9zZGL9sbOblRVr9QrylmSlG0n/CBG2RPebpwt/n0dndCcze0nTuFyqVWtT9G0XSQt7tbSRuPSYom2+lH/VxsRU7YTOlr99zo6ER5qIdcjfbJN1S35ltLtohvVNlTKblRfopwlCWlud7CvrAEIgLOkpQAkbV0elxZLRKiFhjYHR0826W2OPLicnQ62pMXdnZIBfj4H44dDdAq4OoTDJCEzhmmLFhXh9SlaF6WkMxl9iXKWJGRH0SmcLjcZcRGkx/l5erTmLJXvBnuzf19LB0ItZiZnqVWtz6nYA/ZGsMZ2pnIlo1O6w8/z7kwm6euWpnucJVXk7WO0Ra4acXJOlLMkIZ0jTgIwlDQuC2IzwO3s7GySjOnDVOuyz9Fq3LJmgVm+AalN7Q4KykV0d4a/OuG6oqXDJXeWDlc1Uddi19kaSXA5oXir2FbO0jlRzpKEaMXIM4cHSDPDm4qTs8hb+7JTzpIP8TpLchaV7iw6hcstxuak2sL9/4Le2sHN4JJPE2xItJWcpChAnYc+o6pARHfDYiBlgt7WBD3KWZKMDqeLncUBqpXQ8K5q5XSWpg6NA+B4dTM1Te36GiMLWr2SJqooGX4bYH02UidBaCS01XWqMUuGVrek0uE+QkvBZc6QMrrra5SzJBn7yhpo63ARFxnKyKTowLyo9oVXskXKVW1cZBijksXfUq1qfUB9KdQXC40ubWSOZGwLVL2ShiW082+pqaJLhhbh1f62igGiLW5VCq5PKGdJMrQv8+lD4zGbA6Rdk5ILoVFigvzJA4F5zQDjrVsqUs7SgCnxRJVSJoI1QA59AOlwusgvrgMCVK+k0TUVJyGa47mrpJ52h1NnayRAiyypTrg+oZwlydjhcZamDQvQihbAEgKZnlWtpBdqr7OkunEGjjcFJ+dFuqC8gRa7k9jwEG9EMiB40+FyRpaGJ0YxJCoMu8PF3tIGvc0xNg3lUFfkie7O0NsaQ6CcJcnY4Yl8TPOnYnBPSF7krTlLu0vVqnbAFMsd/tfqlWZkJwQuuguQORMwwalCaKwM3OsGCJPJ5F0EKr2lAaKdg8kTIDxWX1sMgnKWJKKsrpXy+jYsZpNXGyhgZMndujw8MYoEtaodOB2tUL5LbEta3K19kU8PZHQXINzWZfyQnOehJoeyVUV4B0axEqPsL8pZkggtqjQuLYbIMD+OV+iJLG1VexyaqgL72gHAZDJ5o3U7VJH3+VO2E1wOiE6FuKF6W+Nz3G53Z2Qp0M4SdH75Sbpome6pAduhhuoODMmlO/yBcpYkomtxd8AJt0HyeLEtaSquc/q5SgGcN13FKCUcnltyqpWTje2EmE1MzooLvAFZctctTcyIJSzETE2znePV8k0MCAj2ls7oroos9RnlLEmELsXdXdHSKtKuarV6iTq1qj1fJG9X1qK7E9JjCQ/VQbtG+/Kr2CPlUF1riIXJ2lBdlYo7P8p2iOhuTBrYsvS2xjAoZ0kS2jqc3uG5AS/u1pC8dTk3w0aYxUx1UztFtS16m2M83G7pi7t3FtUBMFWvc9CWJVKcLgeU5etjg5+ZrhT1B0ZRl3lwEkZ3/YVyliRhd0k9Dpeb5BgrmfF+Hp57NrQvwLJ86GjTxwY/Eh5qYWKG6BxRF+rzoOYotNaCxQppk/S2xi94u1H1iu6aTEKRGaBkqz42+JlpHkV9bVKBop9o0h1DVb1Sf1DOkiR0lQww6bVaiM+G6BRwdYhCXgmZrkYunD9aVCl9KoRY9bXFD7R1ONnvje7G6WdI5kxxr4l/SoYWtTtc1URDW4fO1hgMl0v66K6/UM6SJHiLu/Va0YJY1Wp1S5K2LntTAKpeov90Le6WkK7R3Yw4naK70OUc3CpSn5KRFGMlKyECtxt2F9frbY6xqD4k5geGRkJqrt7WGArlLEmA2+1mpzf8H6evMdpqpWSbvnb4Cc0ZPVTVSH2rWtX2C8mVu4MiuguQNgXMIdBUAfUl+tnhR6ZmeWQ81Pih/qGlZtOniXmCij6jnCUJKKptobrJTpjFzIT0AItRnk6mNlRX3lXt0IRI3G68878UfaC1Dk4WiG1JI0ud3ahx+hoSFtkpTil73ZJylvpHqWcRm6lGnPQX5SxJgLa6mpihU7tyV9ImgTkUmirFZHkJ0S7U+Z7OJ0Uf0CKNCTkQnayvLX7A7Xazw/N50K0btSveuiU5nSWtbmlnsZLx6Bclylk6X5SzJAFavVJQXKRDIzpz4dJfqNWqts9IXlRacqqV6qZ2Qi0mJmboHN2FzuidpOfguLRYrCFm6lo6lDhlX2lvgqr9YlsNz+03ylmSgB0n6gCdi7u74l3Vylm3NNWbAlCr2j4jeXG3Ft0dn27TP7oLnZGD8l3gaNfXFj8QFmIm1+OU7lQR3r5Rng9uF8RmQmya3tYYDsM5S08//TTZ2dmEh4eTl5fHli1nb4999tlnufDCC4mPjyc+Pp758+efsf8tt9yCyWTqdlu4cKG/34bPaGp3cKDC064cdM6SnKvasaliVVvfqla1fcLlhNIdYlv7bEiGt15JT8mArsQPh8hEcNqhfLfe1viFqUpvqX9o1+PM6fraYVAM5Sy98cYb3HPPPTz00EPs2LGDyZMns2DBAqqqeh7cunbtWr7xjW/w6aefsnHjRrKysrjiiisoLS3ttt/ChQspLy/33l577bVAvB2fsLu4DpcbMuIiSIkN19scgVrVKrpy8iDYGyE0qnN+oGQEVb0SeMQp5V60TPUOtq7T1xCjoEX6VQruvDCUs/TEE0+wbNkybr31VsaPH88zzzxDZGQkL7zwQo/7v/LKK9x5551MmTKFsWPH8txzz+FyuVizZk23/axWK6mpqd5bfHyQXPD6wHa958H1RHx256q2Yo/e1viFKZ4hqWpV2we0Dpz0qWAOghSVj2m1OykoD7LoLnRR8pZTnFJzTA9UNNBid+hsTZDjdncp7pYzuutvDOMs2e12tm/fzvz5872Pmc1m5s+fz8aNfZuw3dLSQkdHBwkJCd0eX7t2LcnJyYwZM4Y77riDmpqaXo/T3t5OQ0NDt5teaLUS04Ml/A+DalWrIkt9wHuRljP8v7ukDofLTUqslXRbkER3oUuRt5y1g6m2cNJs4bjcQhBU0QsNpUJ3y2SBtMl6W2NIDOMsVVdX43Q6SUlJ6fZ4SkoKFRUVfTrGfffdR3p6ejeHa+HChbz00kusWbOG3/72t3z22WdceeWVOJ3Osx7nsccew2azeW9ZWfpMbna73ez0aP3oNrjzbGir2mI5V7VavcSBikZa7Wf/rCiA0u3iXtIVbdcUnK5ilKeTPg1MZiHh0VCutzV+QTsPlTjlOdAc5pQJQodL0W8M4ywNlN/85je8/vrrvP3224SHd67+brzxRhYtWkRubi6LFy9m5cqVbN26lbVr1571WMuXL6e+vt57Ky7WR0+osKaFupYOwkLMjEuL1cWGsyJ5R1yaLZyUWCtOl5s9pWpVe1YGQbtyV+XuoMIaDckTxLakEd5pKsLbN0pVCm6gGMZZSkxMxGKxUFlZ2e3xyspKUlNTe33u448/zm9+8xs++ugjJk3qfdp5Tk4OiYmJHDly5Kz7WK1WYmNju930IN9TLzMxPZawkCD7V2ZMA0xQXwSNfYv8GQmTyeQduaBUhHuhbKenXTlDynbloBo11BOS1y0pGY8+osQoB0yQfcOenbCwMKZPn96tOFsr1p49e/ZZn/e73/2ORx55hFWrVjFjxrk/KCUlJdTU1JCWFvwXdk1BekpWkK1oAawxnZ1PkkaXul6oFWdBW9FmyFmvVFzbSnWTnVCLSf9RQz0hed3ShHQboRYT1U3tlJxq1duc4MTZAWX5YlvS6G4gMIyzBHDPPffw7LPPsmLFCgoKCrjjjjtobm7m1ltvBWDJkiUsX77cu/9vf/tbfvGLX/DCCy+QnZ1NRUUFFRUVNDU1AdDU1MS9997Lpk2bKCwsZM2aNVx77bWMHDmSBQsW6PIe+0O+t14pTlc7zop3VStnCsDbulx0Sq1qz4bkK1otBTchWMQoT0dLu5TtBIddX1v8QHiohfEeJ1XVLZ2Fqv3gaAWrDYaM1Nsaw2IoZ+mGG27g8ccf58EHH2TKlCnk5+ezatUqb9F3UVER5eWdhYx/+ctfsNvtfO1rXyMtLc17e/zxxwGwWCzs3r2bRYsWMXr0aJYuXcr06dNZv349VqtVl/fYV9o6nOz3tCtrbexBh+R1S7kZNixmE1WN7ZTXt+ltTnAieXG3loIL2gXLkJEQHgeONqjcq7c1fmGqJuOhIrw901WM0myor/ygIkRvA/rLXXfdxV133dXj704vyi4sLOz1WBEREXz44Yc+siyw7CtroMPpJjE6jMz4CL3N6RnvqnYHOB1gMdzHrVciwiyMS4thb2kDO4vqSI8L0v+DXtSXQmO5p115it7W+AUtuhu0CxZNxuPIarFoyZimt0U+Z+rQOF7coGoHz0qJZ8GiUnADQrmZBqXrRTqo2pW7kjgarLHQ0dLZESUZqsi7F7QVbcp4KduVu0Z3g64TriveCK+cRd7a335fWQNtHUrG4wxK5U6FBwrlLBmU/GDVV+qK2dxZ2Ctt3VIcgFfvStGFUrnHK+wvF9HdIVFBHN0FyJJbIDYzPoLEaCsOl5t9ZUrGoxutp6D6kNiW9DwMFMpZMihaJCNow/8aktctac7qntJ67A6XztYEGSVy1yt1dqMGcXQXPAsWE5wqhKaTelvjc0wmU6c4pZoT1x1tgHX8cIgaoq8tBkc5SwZEa5M1mWBSZhC2K3dF8rEn2UMiiYsMxe5weeeDKRA1auX5YlvS8P/OYK9X0gi3QdJYsS3peegVp1SzGrvjbbCQ8xwMJMpZMiDainZUcjQx4aH6GnMutJO05jC01Oprix8Q4pRxgKpb6kbVflGrZrXBkFF6W+MXNFHYKcHaCdeVQSROqeiCtxNOzuhuIFHOkgEJ+g6crkQmQMIIsa2FhCXDO1RX1S11ol2kM6ZK2a5c09ROca2I7k42wnkouTjlpEwh41Fe30Z5vRKnBMDt7vx/q3qlASPfVWwQoIWag1K5uyckT8Vpq9p85Sx1Uip3u7L2vx6RFE1ssEd3ofMcLN0uUqSSERkWwtjUGEBFl7ycOg6ttWAJg9SJeltjeJSzZDBcLje7i0XHhyEiSyC9kvekzDgATtS0UNPUrq8xwUKJ3IM7d3Yp7jYEiWPkl/HwpuJUOhzoPAfTJkNIcIssGwHlLBmMoyebaGx3EBlmYXRKtN7m9A3vqnYbuOTrGLNFhDIyWfwvVHQJaKvvbFeWtLDUUKlwOE3GQ9K6pSxt/FCdvoYECyoF51OUs2QwtBVtboaNEItB/n0pEyAkQnyJ1hzR2xq/oEYudKF0B+CGuGEQlai3NT7H5XKzy2jOEnQuWorljPBOG6ZkPLqhxCh9ikG+bRUa3nZlI3TgaFhCIX2q2JY0FTdVtS53Ivnw3GPVIrobHmr21skYgq4RXgnpKuOxf7DLeHS0QflusS3peRholLNkMLzK3UZa0YL0KsJavcSu4nqcLre+xuiN5MrdWvRwUkaccaK70JmGqzkilJ0lo6uMR/5gr1uq2AOuDohMFBFexYAx0JmuaLE7OFghVkxBPeakJyRX8h6dEkNkmIWmdgdHqpr0Nkc/urYry1rcbcToLggF54Qcsa11K0qG1iE86GU8uqbgglld3kAoZ8lA7C6px+WGNFs4KbHhepvTP7QoQ9U+aJfPmbCYTUz2dMUN6m6cuhPQUg3mUEjN1dsav5BvtE64rmjnYYmczpKS8fDgFaOUM7qrB8pZMhCG68DpSmwa2LLA7YIyWcUp44BBXuStRZVScyHUYA59H2i1OzlY2QgY9DyUXMZDEwg9UdNCbbNdX2P0RHXC+RzlLBkIQ69ooUvrspypOFXkjfTF3XtKRU1aSqyVNJsBnUHt/1K6XaRMJcMWEUpOUhSAt2Nx0NF0UkR4MUHGNL2tkQblLBkIb3G30eqVNLpeqCVEc2IPVzXR2NahrzF6IXlxt3ceXFYcJiPWgqTkgsUqlJ1rj+ltjV/Q9JYGbTpcOweTxoghygqfoJwlg1Be30pFQxsWs4ncDIOeABldUgASrmqTYqxkxkfgdov6skGHwy59u3KncrdBFywhYZA2SWzLumjR0uGDNbKkUnB+QTlLBkFLwY1JiSEizKKvMedL2mQwWaCpEupL9LbGL2jRpUFZYFq5B5ztEJHQ2XUlGYauG9TIkLtuSZMP2FVch2swynh4O+Gm62uHZChnySDkG7VduSthkZ0DHSUVxpsymJW8vSva6VK2K1c2tFFe34bZJKbcGxZvkbec5+CY1BisIWYa2hwcq27W25zA4nJ5FPSRVrpDL5SzZBC0L1/DiVGeTobcF+rO1uVTuCVMNfaK5MXd2jk4OiWGKGuIvsYMBO3/U7FHKD1LRqjF7HVmB12Et/oQtDdAaCQkjdPbGqlQzpIBcDhd7CkVNTBTjRxZAumLvCek2wi1mKhuslNyqlVvcwKL5LOotC5Hw5+DccOEsrOrQzhMEtKZDh9kRd7aOZg+FSwGduiDEOUsGYCDlY20djiJCQ8hJzFab3MGhhZZKssHp3wdY+GhFsalxQKDbFXb0qW7KkPOWgnDS3domEzS6y1pBfiD6hwEJUbpR5SzZAC0E35yZhxms8FrQYaMBKsNHK1QtV9va/zCoCzy1iKFQ0ZChEE7xXrB6XJ7o7uG7YTrijfCK3c6vKC8kVa7U19jAommzK464XyOcpYMgLdeyejhfwCzuVMoTdpVbRwwyJwl7X8p6UX6UGUjLXYnUWEWRiYbPLoL0tcOptnCSY6x4nS52Vs2SGQ87M1inBSoyJIfUM6SAZCiXbkrmXLPp9L+T3tL67E7XPoaEygkL+72Rnez4rAYPboLngWLSSg9N53U2xqfYzKZOhctg6UztWynGCcVmwGx6XpbIx3KWQpyGto6OHpSDJ6Vx1nytLRKmgIYnhiFLSKUdoeLAxUNepvjf9zuzjScpM6SpgYtzTkYboPE0WJb0vOwU5xykBR5d5XuUPgc5SwFObuL63G7ISshgiHRVr3N8Q3ayVx9CFrrdDXFH3Rb1Q6GVFzNUWirg5BwSJmotzV+QbroLnQuWiRNxWljTwZNZEnyblS9Uc5SkKOtaKfKUFSqEZUI8dliu2yHrqb4i0GVAtAu0mmTwRKqry1+oLGtg8NVnuiuDHWDGprCs6SRpUmZNswmKKtvo7JBPj2pM/CmwpUYpT9QzlKQI+WKFroUmEpat+QVp6zT1Y6AIHlx954SEd3NiIsgOSZcb3N8h/b/Kt0hlJ8lI8oawuiUGGAQKOrXl0JjuRgnlTZFb2ukRDlLQYzb7ZZjzElPSN66PCUzDoBj1c3Utdj1NcbflMg9i2qnrAuW5PFC6bm9QaTEJWTqYFm0aNfRlPFirJTC5/TZWSorK/OnHX3m6aefJjs7m/DwcPLy8tiyZUuv+//zn/9k7NixhIeHk5uby3/+859uv3e73Tz44IOkpaURERHB/PnzOXz4sD/fQp8pOdVKTbOdUIuJ8R6hQ2noOsxTwrEg8VFhZA8RFy2pL9QdrVC5V2xLGv6XNrprCRFKzyDvomWwKHl7xSjlPAeDgT47SxMmTODVV1/1py3n5I033uCee+7hoYceYseOHUyePJkFCxZQVVXV4/4bNmzgG9/4BkuXLmXnzp0sXryYxYsXs3fvXu8+v/vd73jqqad45pln2Lx5M1FRUSxYsIC2Nv1z3Ds89Urj022Eh1p0tsbHpOaCORRaauBUod7W+IWpQweBinD5bnA5ICoZbFl6W+Nz3G63XDpnp6M1W0ha5K0JiO4uqcfpkm9R5kWJUfqdPjtLv/71r/nud7/L9ddfT21trT9tOitPPPEEy5Yt49Zbb2X8+PE888wzREZG8sILL/S4/x//+EcWLlzIvffey7hx43jkkUeYNm0af/7znwFxIXzyySd54IEHuPbaa5k0aRIvvfQSZWVlvPPOOwF8Zz2jfckafnhuT4SGC4cJpJ0TNyg64rp24Jgk0B86jdK6Vqqb2gkxm5iYYdPbHN+TKbc45cjkaKLCLLTYnRyqbNTbHP/gdAiNJZC2E+5ARQP1rfqOx+qzs3TnnXeye/duampqGD9+PO+//74/7ToDu93O9u3bmT9/vvcxs9nM/Pnz2bhxY4/P2bhxY7f9ARYsWODd//jx41RUVHTbx2azkZeXd9ZjArS3t9PQ0NDt5g+kDf9rSN66rP3fdhXX4ZYw1Qh0Ke6Ws15JOwfHpsXIF92FznOwap9QgJYMi9nEZNkXLVX7xPgoqw2GjNLbGr+w7KVtTP7VR2wt1CdQA/0s8B4+fDiffPIJDzzwAF/96leZNGkS06ZN63bzF9XV1TidTlJSUro9npKSQkVFRY/Pqaio6HV/7b4/xwR47LHHsNls3ltWlu/TD263m/YO0aEir7Mkd5H3uLRYwkLMnGrp4ERNi97m+Act/C9prYQ0w3PPRmw6xKQL5eeyfL2t8QvSy3h4xSiniXFSklHT1E5xbSsmE4xJjdHNjpD+PuHEiRO89dZbxMfHc+211xIS0u9DGJ7ly5dzzz33eH9uaGjwucNkMpn4zw8vpL6lg9gISf/GWjSifDc47BASpq89PiYsxMyE9Fh2FtWxs/gU2YlRepvkW5qqoL4IMHUWCktGZ3RXIp2z08mcDgVlYtGSPVdva3yO5ixJq+QtuXq+dg6OSIomNlw/Hbd+fQs/++yz/PjHP2b+/Pns27ePpKQkf9l1BomJiVgsFiorK7s9XllZSWpqao/PSU1N7XV/7b6yspK0tLRu+0yZMuWstlitVqzWwKhp2yLlE/nzkpADEQnQWguVe6RM5UzJimNnUR35RXV8ZWqm3ub4Fm1FmzQWwiXr1gQ6nC72lIohrNJGlkAUBRe8L+9ga09h/uGqJhrbOojR8QvXL0jeCRcs5Sh9jtktXLiQ++67jz//+c+89dZbAXWUAMLCwpg+fTpr1qzxPuZyuVizZg2zZ8/u8TmzZ8/utj/A6tWrvfsPHz6c1NTUbvs0NDSwefPmsx5T4UNMJum7caTuiCuVW1/pYEUj7Q4XMeEh5MgWFeyKt3ZQzkaL5JhwMuIicLuFwKhUtNZ1amRJuNgEAzpLTqeT3bt3s2TJEn/a0yv33HMPzz77LCtWrKCgoIA77riD5uZmbr31VgCWLFnC8uXLvfv/8Ic/ZNWqVfz+97/nwIED/PKXv2Tbtm3cddddgEh13X333fzP//wP7733Hnv27GHJkiWkp6ezePFiPd7i4EPybhytk3F/eQNtHU59jfE1kit3dxWjNJvl6/Tzkj5FKD83lkFDcOjp+ZrOobp1utrhc7RxUfHZYoyUZLhc7qBxlvqchlu9erU/7egTN9xwAydPnuTBBx+koqKCKVOmsGrVKm+BdlFREeYuBW5z5szh1Vdf5YEHHuBnP/sZo0aN4p133mHixM5hnz/96U9pbm7m9ttvp66ujnnz5rFq1SrCwyUaaxDMZMhd5J0ZH8GQqDBqmu3sL29g2lBJal9cTijV2pUlDf/LXtytERYl1Lwr94hFy/hFelvkc6ZmxfHB7nL5xp5IPg/ueE0zjW0OwkPNuhZ3w3kUeOvNXXfd5Y0Mnc7atWvPeOz666/n+uuvP+vxTCYTDz/8MA8//LCvTFT0hwxPB2XtMWiphcgEfe3xMSaTiSlZcaw5UMXOojp5nKXqQ2BvhNAoSB6ntzV+QVN9lt5ZAhHhrdwjooUyOktdxp643W5MsmiCeTvh5IzuaguWiek2Qi36dvrJ12eoMBaRCTBkpNhW4pTGQbtIp08Fs3z6Q/WtHRw9KXSHBo2zBNKegxPSbYSYTVQ3tVNa16q3Ob7B7e5S3C2psxQkKThQzpIiGMiQvG7JW+QtUeuy5MXdu0vqAMhKiGBIdGA6X3VFOwfLdgpFaMkID7UwzjNfU5pU3KnjopPYEtY5DUEygmmQvHKWFPojuTjlpCwbJhMU17ZS09Sutzm+YZCE/6fKrK/UlcTRYI2FjhY4WaC3NX6haypOCrTuxdRJECKfQ9/W4aSgXEzHUJElhQK6ywdIOBYkNjyUEUnRgCQX6vYmqNovtiUtLA2m8H9AMJs76wdl1VuSLR0ueQpuX1k9DpebxGgrGXERepujnCVFEJAyESxWaKuDmqN6W+MXpLpQl+eL8RixGRCbds7djYbb7e6UDQiC8H/A8KbD5axb0s7BPaX12B0ufY3xBaVyd8LlF2uCsLagKMhXzpJCf0LCIG2y2JY0FecduSBDvYQ3BSdnvVJxbSu1zXZCLSbGp8mnTH5WJE+HD0+MwhYRit3h4kCFf4afBwxHO1TsEduSnofBFt1VzpIiOPCqCMt5odZO+F3FdbhcBk81ele0cob/tRli49NiCQ+Vr9PvrGiRpZMHoU0ypWs6ZTxAgghv+W5w2iFyiBCklJBO6Y7gqBtUzpIiONC6qiRd1Y5NjSE81Exju4Nj1U16mzMwZC/uDrIVbcCIToK4YYAbSnfobY1f8DpLRo/wdk3BBUGKytfUNLVTXNuKySQaZIIB5SwpggPti7diL3S06WuLHwixmJmUEQcYPBVXXwqN5WI8RvoUva3xC8HUrhxwJE/FSTP2ZJAsWEYkRRMbJIOPlbOkCA7ihkJUErg6oGK33tb4hSkytC5rX6Ip48WYDMmwO1zsK9PalYMj/B9QZC/yzowD4Hh1M3Utdn2NGQjeTjg565V2BWF0VzlLiuDAZOpyoZa7ddnQkSXJV7QF5Q3YHS7iIkPJHhKptzmBJ7PLOSihjEd8VBjDE4WTb9hFS3M11J0ATNIWd2uRv8nKWVIoeiCzi96ShGjO0sHKRlrtTn2NOV8kH9ypfYFOzowLinblgJM6Ccyh0KJ9IcuH4Rct2jmYOBrCg6Oex5e4XG5vZGmqcpYUih7IkLteIs0WTkqsFafLzZ5SA3YbOTvEOAyQ3lkKpvB/QAkN7xydIfmixbCRJcnFKI/XNNPQ5sAaYmZMaoze5nhRzpIieMiYBpigrgiaTuptjc/p3rpswDlxlfvA0SpWs9rwY8kY1MXdGpIP1dXGnuwqqcNtxFSj5NIdWqdiboaNUEvwuCjBY4lCEW6DpDFiW9LoklY0bMhVrbaizZghxmNIRl2LnePVzUBnIfCgxKt5Jmft4NjUWMJCzNS1dFBY06K3Of3D5eqUdZC0bjBYo7vyXfEUxsZb5C2rsxQHGLReYpDUKw1PjCI+KkxfY/REKxou3w0OA3eMnYWwEDMT04Uy+84ig0V4qw9BewOERkLyeL2t8QuaKOzUocHVjaqcJUVwIbk45aRMG2YTlNe3UdlgMD0p6WdR1QHBt6INOAk5EJEAznao3KO3NX5B+yI2XIRXi/alTwNLiL62+IFWu5OC8kagM10aLChnSRFceIu8d4iQs2REWUMYnSKKFg0VXWqphZojYlubTi8ZylnyYOrSki6r3pJRi7wlL+7eU1qP0+UmNTac9LgIvc3phnKWFMFF8ngRYm5vECFnCZlqRHFKrdh3yEiITNDXFj/gdruVs9QVyeuWtP/x/rIG2joMJOMheSp8R5GWgovT15AeUM6SIriwhEDaFLEtaSrOkB1x3hWtnBfpwpoW6lo6CAsxMy4tVm9z9EfydHhmfASJ0WE4XG72lRlExqOtAar2i21JI0s7lbOkUPQD6cUpRb3E7hIRcjYEkof/Ncd1QrrolBr0aGm42mMiBSsZQsZDnIeGSYeX7QDcYBsKMal6W+Nz3G43Ozz/i2Ar7gblLCmCES16IemqdmRyNFFhFlrsTg5VNuptzrlxuTprVySNLGnaLioF5yEivlNLS3K9JcMM1dUWLFlynoNl9W2cbGwnxGxiYnrwKZMrZ0kRfGhF3pX7wW4wHZQ+YDGbmOTR8TFE3VLNEWivh5AISJ6gtzV+QdUr9cAgqVvKN0pkSfJ6JS0FNy4tlogwi87WnIlylhTBhy0DYtLA7YTyfL2t8QveIm8jXKi9YpRytiu3dTjZX94AwNSs4Av/60aG3OnwSZk2TCYorWvlZGO73ub0jtstfd3gTm8KLk5XO86GcpYUwYn3Qi35qtYIkSXJ65X2lzfQ4XSTEBVGVkJwtSvrStexJ0YcC3IOYsJDGZUcDRjgPKw9Bi01YAnrnN0nGVpkaVoQ1iuBcpYUwUqm5ErentXToapGmtod+hpzLrT/gazjFbrUK5lMJn2NCSZSJkJIOLTVQc1Rva3xC52K+kHemaqdg2mTIcSqry1+oN3hZG+pJ7qrIksKRT/IkHuYZ3JMOBlxEbjdsDuYV7XtTVC1T2xLGv7XogpTVb1SdyyhnTIe0kZ4DaLk7Y3uztLXDj+xv6wBu9NFQlQYQxMi9TanR5SzpAhO0qeCyQwNpdBQrrc1fsG7qg3mC3XZTnC7IDYTYtP0tsYveIu7g3RFqyveVJycEV4tihH0Mh6Sp8K99UpBHN1VzpIiOLFGdw6KlPRCbYi6Jckv0jVN7RTVio5LrUNR0QXJi7xHp8QQGWahqd3Bkaomvc3pGXsLVO4V25JGd7UFY7Cm4EA5S4pgRvILddexJ+5gLaAtlVtfaVdJHQAjkqKwRYTqa0wwov3fK/dCR6u+tvgBi9lEbobQ9AlaRf3yfHA5IDoVbJl6W+MXOpW7g7O4G5SzpAhmMuWuW5qYYSPEbOJkYztl9W16m3Mmg6BdeceJOiC4L9K6YsuE6BTxZV2+S29r/IL2vw/aCG/X6G6QpqgGQlVjGyWnWjGZhJxDsGIYZ6m2tpabbrqJ2NhY4uLiWLp0KU1NZw+b1tbW8oMf/IAxY8YQERHB0KFD+X//7/9RX999DpDJZDrj9vrrr/v77Sj6grfIewe4DDTsso+Eh1oYmxYDBGk3Tn0xNFWCORTSJultjV/YEeTtyrpjMnWeh5JGeDs74up0teOseJW75Szu1rpRx6TEEBMevNFdwzhLN910E/v27WP16tWsXLmSdevWcfvtt591/7KyMsrKynj88cfZu3cvL774IqtWrWLp0qVn7Pv3v/+d8vJy723x4sV+fCeKPpM0BsKioaMZqgr0tsYvBLWKsHaRTp0IofLpDzldbnZ5ognThsXpaktQI/lQXS0dfqiykeZgk/Fwu6FY8uhukItRahhCjregoIBVq1axdetWZswQq5w//elPXHXVVTz++OOkp6ef8ZyJEyfy73//2/vziBEj+PWvf83NN9+Mw+EgJKTzrcfFxZGaKt9gQsNjtoiuuML14kKdOlFvi3zOlKx4Xt5UFJwpAMnHKxysaKTZ7iTaGsKo5Bi9zQlevGNP5HSWUmLDSbOFU17fxu6SemaPGKK3SZ3Ul0BTBZgsnTIOkuGtVwpy9XxDRJY2btxIXFyc11ECmD9/Pmazmc2bN/f5OPX19cTGxnZzlAC+//3vk5iYyKxZs3jhhRfOWWzb3t5OQ0NDt5vCT0guTqmtpvaU1tPhdOlrzOnIXq/kuUhPyYrDYpavFsRnpE8FTCIt21iptzV+oWuzRVDRNbobFpz6QwPB4XSxu0SUxgR7ZMkQzlJFRQXJycndHgsJCSEhIYGKioo+HaO6uppHHnnkjNTdww8/zJtvvsnq1au57rrruPPOO/nTn/7U67Eee+wxbDab95aVldW/N6ToO5KLUw4fEkVseAjtDhcHKxr1NqcTRzuU7xbbksoGdNYrxelrSLBjjYHkcWJb0lRc0Cp5e6O7ctYrHaxspLXDSYw1hBFJ0Xqb0yu6Okv3339/jwXWXW8HDhwY8Os0NDRw9dVXM378eH75y192+90vfvEL5s6dy9SpU7nvvvv46U9/yv/+7//2erzly5dTX1/vvRUXFw/YRsVZ0KIaVQXQJl8Ez2w2MTkYL9QVe8HZDpFDIH643tb4Ba8Q3rDgDv8HBZLLeHRV8g4qGY+SLeJe0uiudg5OGRqHOciju7o6Sz/+8Y8pKCjo9ZaTk0NqaipVVVXdnutwOKitrT1nrVFjYyMLFy4kJiaGt99+m9DQ3qvt8/LyKCkpob397FOorVYrsbGx3W4KPxGTAnHDALe0q9qpwdiN0zUFJ2G7cm2znePVzQBMC/JaiaDAW7ck59iT3AwbFrOJqsZ2yoNFxsPR3inXIGl0t6tyd7Cja4F3UlISSUlJ59xv9uzZ1NXVsX37dqZPFyucTz75BJfLRV5e3lmf19DQwIIFC7Barbz33nuEh4ef87Xy8/OJj4/HapVvWKFhycqDuhNQvAVGXKa3NT5nmieysSOYIkuSK3drUbwRSVHYIoO3XTlo0D4HZTuFjIfZoq89PiYizMLY1Bj2lTWws6iO9Lgg6P6s2ANOO0QkQEKO3tb4hZ0eIVAjRHcNUbM0btw4Fi5cyLJly9iyZQtffPEFd911FzfeeKO3E660tJSxY8eyZYsIWzY0NHDFFVfQ3NzM888/T0NDAxUVFVRUVOB0Cs2e999/n+eee469e/dy5MgR/vKXv/Doo4/ygx/8QLf3qugBTV+keIu+dvgJTRSvsKaF6qazRzQDiuYsZcjpLCl9pX6SNFbIeNib4ORBva3xC53jh4Jk0SJ5dLeuxc6xkyK6O8UAo4YM4SwBvPLKK4wdO5bLL7+cq666innz5vG3v/3N+/uOjg4OHjxIS4uY87Rjxw42b97Mnj17GDlyJGlpad6bVmMUGhrK008/zezZs5kyZQp//etfeeKJJ3jooYd0eY+Ks6A5SyVbwRVkHWM+wBYRyugUUdy440QQXKibqkQkDxNkTNPbGr+glLv7iSbjAdKmw4NuVqO2OMyStF7J83fOSYwiPipMX2P6gCF0lgASEhJ49dVXz/r77OzsboV5l1xyyTkL9RYuXMjChQt9ZqPCTyRPgNAoaG+AkwcgZbzeFvmcaUPjOVTZxI6iOq6YoLPml7aiTRoL4cE7fuB8cThd3plwSoyyH2TOEJpnxVtg2hK9rfE5muOsyXiEWnSOJUiuc9a1uNsIGCaypBjEWEI6IxzFfdfVMhLeuqVgiCxpf2NJxyscrGykRYlR9p8sT32opOnwnEQh49HW4eJAuc4yHo0VUF8EmCBdzuiuEYbndkU5SwpjoF2oJe3Gme5xlnaV1GF36JxqLPI4S0Mv0NcOP6GNV1BilP1E0/qpPggttfra4gfMZpP3i3v7CZ3fnxZVSh4H4fJ1W7tcbm+60widcKCcJYVR8K5q5Yws5SRGERcZSrvDxf5yHfWkHO2i4wk6/+aSsfOEEqM8L6KGwJBRYlvyRct2vWU8JO9GPVbdRGObg4hQ0YVoBJSzpDAG2kWj5gg01+hrix8wmUxM965qdUzFle/2iFEmStuurHXCGaFdOeiQfNEyQ3OWCvWOLGnOkpypcO0al5tpI0Tv2rA+YgwrFYrIBEgcLbZL5KyZCIq6peJN4j4rT8p25ZqmdgprRMesEqM8D4Z6nKUiOZ2lyZ7UbFl9G2V1rfoY4eyA0h1iW9LI0rZCcY2bYaAFi3KWFMZBcr2l6cEgTil5cbfWgaPEKM8TLbJUul18qUtGlDWEcWkiLaRbhLdiNzhaRSdq4hh9bPAz2t92RrZylhQK3yN5N87kTLGqLddrVet2d9F2kbNeSYlRDpAhoyAiXnyZV+zR2xq/oHs6XIvaZeWBWb6v6Jqmdo5po4YMdB7K959QyIuWv5d0VRsRZmF8muh80eVCfaoQmirBHNopQCgZXmfJQOH/oMJs7jwPJa1bmp6dAOjpLG0U95IuWLS/66jkaOIig1+MUkM5SwrjkDhahKZlXtUO03FVq0WV0qdA6LnnKBqNDqeLXcX1gLFWtEFHluTOkucc3F/eQIvdEdgXd7s7/65DZwf2tQOEEVNwoJwlhZHouqqVtHVZ16G6XYu7JWR/WQOtHU5sEaGMSo7W2xzjoulvFW0WX+6SkREXQZotHGcXLaCA0TW6K+mooW0eZ2n6sASdLekfyllSGAvJW5e9q9qyBlrtzsC+uLdeSc7i7s6LdDxmJUZ5/qRPA5MFGsugvkRva/yCbp2p2nUtbTKERgT2tQNAW4eTPSUiumukTjhQzpLCaGhDJSUt8k63hZMaG47D5Wa3Z35ZQGhrgMp9YlvSyNI2j3aO0cL/QUdYJKRNEtuSLlq0L/JtgXaWijzRXUnV8/eV1WN3uhgSFcawIZF6m9MvlLOkMBYZ08FkhvpiaCjT2xqfYzKZuqgIB/BCXbIVcEPcMIjReZCvH3C73Wz1aLvMzDZW+D8oyfJ8mUvrLInPyI4Tp3C5AphqLO7SCSchmr7S9GHxmAym46acJYWxsMZAygSxLWl0SZcUgOSSAUW1LVQ3tRNmMZObYdPbHOMjeZH32LQYIkItNLQ5OHKyKTAv2loHVQViW9LI0jaDFneDcpYURkT7QtdC1pLRtSPOHagCWq24e6iczpIWVcrNtBEeatHZGgnQzsGKvdAeIGcigIRazEzxDHgNWGeqFt2NHw7RyYF5zQDidru9C0CjFXeDcpYURkRrqdX0SCRjfFos1hAzp1o6vOJtfsXl7JxyLmlkSdUr+RhbBtiywO0UumcSoi1atNSR35G8Xul4dTM1zXbCQsxMzIjV25x+o5wlhfHQnKWK3dDeqK8tfiAsxMxkz6p26/EADPSs2g/2JgiLgeTx/n89HfCG/w24og1aZB8/lB1gGQ/Z65U85+DkTBvWEONFd5WzpDAetgywDQW3S9oL9SxPEfKWQEw/11a0mTPAbLyL2LmobbZzpEqkiqYbrF05qPEWecuZDtcGLR+vbqa6qd2/L+bs6IzuShpZ2l5o3BQcKGdJYVSGaak4OS/UM4eLC8rWQDhLkhd3azUnI5OjSYgyzniFoMcbWdoKLpe+tvgBW2Qoo1OEeKnfmy0Gw/Dcos5OOCOinCWFMZG8bmna0DjMJiiubaWivs2/L6Y5nLKKUWr1Sga9SActKRMhLBra6+Fkgd7W+IWAjR/yDs+9QMrhuXUtxo/uyvdfUQwOhs0R9yVbwWHX1xY/EBMeyvh0UQTp11RcXRHUFwlFZkkjS53tysYM/wctlhDI9IjEntigry1+QksZ+T3CK3k3quZs5iRFGTa6q5wlhTFJHA0RCeBog/J8va3xC5p4ol+LvE94InNpk8Eq37y0tg6nVwl9puqE8z3D5or7E1/oa4efyPOkw/eU1vtv/JDb3T2yJCGdDRbGPQeVs6QwJiaT9Kk4rcjbr6ta7UtOi9RJxu6SejqcbhKjrQxNMNZ4BUOgfW5ObJByqG5mvBiq2+F0s7PYT6m4uhPQVCH18FytuNvI3ajKWVIYF63I+4SczpKWNjpY2Uh9S4d/XkRzNLUIgWRsOyEczZnZxhuvYAgypoMlDJoqofaY3tb4HJPJxCxPdGmLvyK8RfIPz80vrgOMrXOmnCWFcRnqWdUWbZSyGycpxkpOYhRud+eXvk9pOgnVh8S2pO3KmqCgqlfyE6HhwmECaeuW/O4sadFdSc/B/OI67E4XSTFWhidG6W3OeaOcJYVxSZsEoZHQVgcnD+htjV+Y6U+9pSLPl1vyeIiUz5lwudyqEy4QdE3FSYhWt7Sj6BR2hx8WZZqzlD3P98cOAjQnM294gqGju8pZUhgXS6gQUgRp65a8ekv+WNVq6UtJ65WOnGyioc1BRKjF21mo8APa56dITmdpRJLQ52rrcLGntN63B2+sgJojgEnayNLm4zVAp9NpVJSzpDA2XVNxEqIVee8praetw8fdON7w/2zfHjdI0Fa0U7LiCLWoS53fyJwFJjOcKoT6Ur2t8Tkmk6lTUd/XixbtHEyZCBHyRT/tDpdXNiAvZ4jO1gwMdQVRGBvJi7yzEiJIibWKbpyiOt8duK0eKvaIbUkjS5uOeVa0OcZe0QY94bGQOklsy7po8dYt1fj2wFrqMlvOBguxyHMRHxnKyCRjS5MoZ0lhbDJnCkHFhhIhsCgZJpOpU2/Jl3VLRZsBN8QPh9h03x03SHC73Wz2RAEuMPiK1hBIrrekOUvbCk/hdPlQIqFQk+6Q01nSUnCzhidgNhu3XgmUs6QwOmFRouUWpJ0TN8sfc+K0+hJJL9LHqps52dhOWIiZKVlxepsjP94ibzkjS+PSYomxhtDY7qCgvME3B22u6RwTI+l5qKUtZw03/oLFMM5SbW0tN910E7GxscTFxbF06VKampp6fc4ll1yCyWTqdvve977XbZ+ioiKuvvpqIiMjSU5O5t5778XhcPjzrSh8jeTdOFpkaceJUzicPurG0f5Ww+SsV9JScFOz4ggPtehszSBAq3s7WSCcAMmwmE1ejSCf1S1pUbikcRBlfGfidBxOl1e6w+jF3WAgZ+mmm25i3759rF69mpUrV7Ju3Tpuv/32cz5v2bJllJeXe2+/+93vvL9zOp1cffXV2O12NmzYwIoVK3jxxRd58MEH/flWFL5GciXvMSkxxIaH0Gx3st8Xq9qOVijdIbYlrVfafEyl4AJK1BBIGiu2JT0PteiIz50lSc/BgvJGmtodxISHMC7N+N2ohnCWCgoKWLVqFc899xx5eXnMmzePP/3pT7z++uuUlZX1+tzIyEhSU1O9t9jYzn/aRx99xP79+3n55ZeZMmUKV155JY888ghPP/00drt8w1mlxbuqPQDN1fra4gfMZpNXVNEnF+qSbeDqgJg0UbMkGW632xtZUs5SABkmeWfq8E7NM7cvRrt49ZXkTMFp9UozsxOwGLxeCQziLG3cuJG4uDhmzJjhfWz+/PmYzWY2b97c63NfeeUVEhMTmThxIsuXL6elpaXbcXNzc0lJSfE+tmDBAhoaGti3b99Zj9ne3k5DQ0O3m0JHooYIYUWAwvX62uInZvrSWfKm4OaIGXuScby6marGdsIsZqYOjdPbnMGD5EXeuRk2wkPN1DbbOXqy9xKQc9J6Cir2iu1hcopRbu4iRikDhnCWKioqSE5O7vZYSEgICQkJVFRUnPV53/zmN3n55Zf59NNPWb58Of/4xz+4+eabux23q6MEeH/u7biPPfYYNpvNe8vKyjqft6XwJdkXivvjcjpLs0eICMnm47W4BtqNI7m+knaRnjJU1SsFFO3zVL4L2hv1tcUPhIWYmTZU1C1tHuiipWgT4IYhIyEm5Zy7Gw2Xy+1tSJmlnKWBc//9959RgH367cCB8x9jcfvtt7NgwQJyc3O56aabeOmll3j77bc5evTogOxevnw59fX13ltxcfGAjqfwAcMvEveSRpYmpscSbQ2hvrVjYHVLzg4o2Sq2Je3A8abgJLlIGwZbBsQNA7cLinuP+BsVn0V4Cz8X95LWKx2sbKSupYPIMAsTM2x6m+MTQvR88R//+Mfccsstve6Tk5NDamoqVVVV3R53OBzU1taSmpra59fLy8sD4MiRI4wYMYLU1FS2bNnSbZ/KykqAXo9rtVqxWq19fl1FAMieC5jEYNjGCojp++fCCIRYzOQNT2DNgSo2Hq05/wtQWT50tAi1YK0gVyJUvZLODJsLdSdEqnfkfL2t8TlaSmnzMVG3dN6zzrzF3XKm4DRncvqweGnU83V9F0lJSYwdO7bXW1hYGLNnz6auro7t27d7n/vJJ5/gcrm8DlBfyM/PByAtLQ2A2bNns2fPnm6O2OrVq4mNjWX8+PG+eZOKwBARD6m5YlvyVNyGowMoYi9cJ+6HzgGzHBexrpyoaaGyQatXkm98RNCjFStLeg5OHRpPiNlERUMbxbWt53eQ9kaRqgTpi7tlqVcCg9QsjRs3joULF7Js2TK2bNnCF198wV133cWNN95IerpQHy4tLWXs2LHeSNHRo0d55JFH2L59O4WFhbz33nssWbKEiy66iEmThDT/FVdcwfjx4/nWt77Frl27+PDDD3nggQf4/ve/ryJHRsSbilunrx1+QnOWthyvpeN89ZaOfSbucy72kVXBhRZVmpxlIyJM1SsFHO0cLN0ObfI1vkSEWZjsETndeOw8Fy1Fm0WqMm4Y2DJ9Z1yQ4Ha7vZElo8+D64ohnCUQXW1jx47l8ssv56qrrmLevHn87W9/8/6+o6ODgwcPervdwsLC+Pjjj7niiisYO3YsP/7xj7nuuut4//33vc+xWCysXLkSi8XC7Nmzufnmm1myZAkPP/xwwN+fwgdoF2pJV7XjUmOJiwyl2e48v+nnHW2dtSTD5XaWVApOJ+KGCjkKt1NaCYG5nkXLF0fOU3zzhKdeKVvOFNzRk81UN9mxhpiZlClHvRLoXLPUHxISEnj11VfP+vvs7Oxu2hdZWVl89tln5zzusGHD+M9//uMTGxU6M3S2mBN36jjUl0i3ajObTVwwfAir9lWw8WiNtzOnz5RsAUcbRKdA0hj/GKkjah5ckJBzMWw/LqKYoxfobY3PmTMykac+OcKGozXnV7dUKLcYpVc9f2gc1hB5oruGiSwpFOckPBbSp4htSaNLc0YKJ2Dj0fNY1WopuOEXSamvVFTbQnl9G6EWU/8dSYXv8EZ4z71YNSJTh8YRHmqmuqmdQ5X91Ftqb4IyTT1fznqlL46I9OTcEYk6W+JblLOkkAuv3pKkdUueiMnWwlraHc7+PVn7m0iegpucGafqlfQk2+MsVe6VUlHfGmLxSghojkGfObEBXA5PujLb98bpjNPlZoNnITd3lHKWFIrgpaveki9GEgQZI5OjSYy20u5wkV9U1/cntjWIoluQtrhbzYMLEqKTIHmC2JZ00TJ3pHAE+t2ZemytuM+5VMro7r6yeupbO4ixhjBJEn0lDeUsKeRi6AVgDoX6YjhVqLc1PsdkMnWREOhHKu7EBlF0G58tVrWS4Xa72eiJLOXlyNOubFg0h1zSVJyWYtp8rBZHfzpTvc7SJT63KRjQit7zcoYQIom+koZc70ahCIuCjOliW1I17zkeZ0lzDvqE5Cm4Y9XNlNe3EWYxM2OYcpZ0R/ucSRpZGp8eiy0ilMZ2B7v72pnaWAlVnpmjkp6HWlpy3kj5orvKWVLIx/DBUbe0s+gUrfY+1i0dl1tfaf2hkwDMyI5X9UrBwLA5ojO19hjUyTcOymI2ec/DDX2tW9LOwdRJYvi3ZLR1ONnimQc3T7J6JVDOkkJGug7VlbBuadiQSNJt4XQ43Ww/cercT2g6KYptobP4VjI+11a0El6kDUl4LGRME9uypuJG9lNvSfIU3PYTp7A7XKTEWhmRFK23OT5HOUsK+ciaBRYrNFVAzRG9rfE5om6pHwWmWjoyeYIovpWMDqfLK6Vw0Sj53p9hkTwVN8dT5L296BRtHeeI8Lrdnc7SiEv9a5hOdJUMOO+ZeUGMcpYU8hEaIRwmkHZV268ib8lTcDuL6mi2O0mICmN8Wqze5ig0tM7UY59JGeHNSYwiNTYcu8PFtsJzRHhrjkBDqVjEDZ0dGAMDjNdZGilndFc5Swo56XqhlhDNWdpTWk9jW0fvO3vFKOV0lj4/LOqV5o5MxGyWb0VrWLLyOiO81Yf0tsbnmEwmr0jsF+eK8B79VNwPzROLOcmob+nwFrorZ0mhMBIjLhP3xz8Dp0NfW/xARlwE2UMicbrcXn2hHqkrEuNfTBZpxyusOyy+qC6U9CJtWELDhXMA0qbiNAmBcxZ5S16vtPFYDW630IFLtYXrbY5fUM6SQk7Sp0K4DdrqoWyn3tb4hQs99TnrPJGVHtG+pDKmiaJbyahv6WB3SR2giruDEi2aqTkLkqFFUfaUCjHGHnE6OusGc2SvV5Kvy09DOUsKOTFbOi/URz/R1xY/caHHOVh/uJdVreT6ShuOVuNyw4ikKNLj5EtvGB4tklL4Obj6OZ7HAKTawhmRFIXL3Tlu5wzKdkJ7A4THQdrkgNoXKGSvVwLlLClkRkvFHftUXzv8xOwRQ7CYTRyvbqa4tuXMHVyuzhX9cDklA9Z7LtIXqi644CRtClhjoa0OyvJ1NsY/eEefnC0V1/UcNMunAVZW18qx6mbMJrhARZYUCgOitegWbxGz0SQjJjyUaUPjgLNElyr3QlMlhEaKMTAS8rlWr6RScMGJJaTTUT+6Rl9b/MScEeeI8GqLNUnrlbSo0uSsOGLDQ3W2xn8oZ0khL/HZkDBCzESTdPSJpiu07lAPdUval9PwiyDEGkCrAsOJmmaKalsItZjU8NxgZuR8cX9EUmdppIjwHuspwtveJBZrIL2zpBW7y4pylhRyo6XiZK1bGi2cpS+OVp850FP7ctK+rCRDW8lPHRpPlDVEZ2sUZ2Xk5eK+ZAu09kFx3mDEhocyfWg8AJ+dvmgp2giuDjG8OiFHB+v8i8vl5nOPgrnM9UqgnCWF7GipuCMfSymMl5thEwM92xzsKuky0LO9EYo2iW3NYZSMz5VkgDGIGwqJo8Htklb37OIxYtFyhrOkLdKGXwwSqlrvK2uguqmdqDAL04fF622OX1HOkkJuhl8E5lA4VQg1R/W2xudYzCbmeZyFbhfq4+vFijZ+OAwZoZN1/sPhdHmFALXomiKI0aKbktYtXez5DG44Uo3d0SXCe3i1uB/1JR2s8j9rD1YBIqoUFiK3OyH3u1MorDGdYoxHVutri5+4xLOq1S5cQOeXkpYCkYzdpfU0tjmwRYSSm2HT2xzFuRjh+RweWSNlhHd8WiyJ0Vaa7c7O4da1x6DmMJhDpK1XWutZoF0yJllnS/yPcpYU8qOt6g5/pK8dfkJLAewuqedkY7t48MjH4l7SeqXPDoqL9ByPfIIiyMmeCyHhYj7ayQN6W+NzzGYTF40WEd61hzyLlsOeczDrAiGQKxl1LXZ2FgnHUFuwyYxylhTyM+oKcV/4Odib9bXFDyTHhHujK58dOgnVR0Ta0RwK2Rfqa5yf+NQTRbt0EKxopSA0AobNFduH5Yzwaqk4zZH3Ls4kTcGtOywEYUenRA8KQVjlLCnkJ3E02IaC0y5qeSTkUs/K7tODVXD4Q/Fg9lywRutolX+oamxjt6eY/ZKx8q9opWH0AnEvaYT3wlFJmExwoKKR8praTrkSSZ0lLe0/GFJwoJwlxWDAZJI+FXfJWHHBWnfoJK6Dq8SDoxfqaJH/WOtZuU/KtJEcI+fQTinRIrwnNkBrna6m+IOEqDCmZsUBcGDjf8DRBrEZkDxeX8P8gMvl9mq7XTJIGiyUs6QYHGgX6sOrpSwwnZwZR3xkqFAqP7FBPKit5CXjkwKVgjMkCcMhcYwQiZW0K+7ycSkAuA52ScFJKBmwp7Se6iY7UWEWZmQn6G1OQFDOkmJwMPxCsFihvgiqCvS2xudYzCYuHp3EhebdmN0OkXqUUATP7nDxuUcx+PJxylkyHJoDf+hDfe3wE+Iz6WZ0w0bxwEg5U3AfF1QCorlEdskAjcHxLhWKsCjIuVhsH/qvvrb4iUvHJnO5Zaf4QdKo0tbCWpraHSRGW5mYLl+HkfRoqeHDH4HLqa8tfmBMSgzzYk+SZarCaQ7rFMWVjI890d3Lx6bobEngUM6SYvAw5kpxf+A/+trhJy4ZNYRLzfkAVCRfrK8xfmKNNwWXhFlJBhiPrDzRRt96Ckq26m2NzzGZTNwyZD8Ah6NmiEWaZJScaqGgvAGzSSzQBgvKWVIMHkZ7nKXSbdBYqa8tfsBWu5sEUyMN7khW1g3V2xyf43a7veF/lYIzKJaQztTUQTkjvLPsYszQWy2TcEtYH6ktWGYMSyAhKkxnawKHcpYUg4fYNEifJrYPrdLXFn9wYCUAn7km8dGBWp2N8T0HKxspqm3BGmLmokHSgSMlWoT3oIQR3sYKYmt2A/B2cy77yhp0Nsj3DNYFi3KWFIOLMVeJe9lWtW43FAhn6UPnTLadqKW22a6zUb7lo33iIn3hqEQiw0J0tkZx3oz6khBMrT4EJw/pbY1v8SzCjlvHcpJ4PtpXobNBvqWxrYNNx2oAmD9+8NQrgYGcpdraWm666SZiY2OJi4tj6dKlNDU1nXX/wsJCTCZTj7d//vOf3v16+v3rr78eiLek0IOxHmfp2Kdgb9HXFl9y8gDUHgVLGOXJF+Jyw5oCuVKNH+0XXzxXTEjV2RLFgAi3iQHX4I2GSoNnEdYyXDRYrJLMWVp3qJoOp5ucxChGJMkneNsbhnGWbrrpJvbt28fq1atZuXIl69at4/bbbz/r/llZWZSXl3e7/epXvyI6Oporr7yy275///vfu+23ePFiP78bhW4kj4e4oUIw7ugnelvjOzxRJXIuYd6E4QCs3i+Ps1Ra18reUlFUevkgKiqVlnFfFvcyOUvtTXBsLQBDZ3+NELOJQ5VNHDt59kW90fjv3nIAvjTIokpgEGepoKCAVatW8dxzz5GXl8e8efP405/+xOuvv05ZWVmPz7FYLKSmpna7vf3223z9618nOrq7RxwXF9dtv/BwpQosLSYTjL1GbBe8p68tvuTA++J+7Je9F7L1h6tptcvRnq2lM2ZkJzAk2qqzNYoBM+YqwASl26Gh52u44Tj8kViExQ8nZmgus0cMAeDDfXIsWto6nHx6QBR3L5w4+KK7hnCWNm7cSFxcHDNmzPA+Nn/+fMxmM5s3b+7TMbZv305+fj5Lly4943ff//73SUxMZNasWbzwwgvn7GBob2+noaGh201hIMZfK+4P/hcc7fra4gvqiqB8F5jMMOYqJqTHkhkfQWuH0zu/yeho9UpXDMIVrZTEpELmTLF94AN9bfEV2uJrwmIwmVjgSRd/KEkqbv3haprtTtJs4UzOjNPbnIBjCGepoqKC5OTuofeQkBASEhKoqOjbB/H5559n3LhxzJkzp9vjDz/8MG+++SarV6/muuuu48477+RPf/pTr8d67LHHsNls3ltWVlb/3pBCXzJnQkwatDd4w+aGRvuyyboAopMwmUxclZsGwAd7ynU0zDfUNLWzpVB0910xfvCtaKVFS8UVvK+vHb7A3gKHPCNOxi0ChGNvMkF+cR3l9a06GucbtBTcggmpg1LjTFdn6f777z9rEbZ2O3DgwIBfp7W1lVdffbXHqNIvfvEL5s6dy9SpU7nvvvv46U9/yv/+7//2erzly5dTX1/vvRUXFw/YRkUAMZthnCcVt/9dfW3xBdp70L58wOssfXKgirYOY6fiVu2rwOlyk5thY+iQSL3NUfiKsZ7Pa+Hn0Fytry0D5ega6GgG21BInwpAcmw404fGA52RUaNid7j42FMDeeUgTMGBzs7Sj3/8YwoKCnq95eTkkJqaSlVV93SCw+GgtraW1NRz/+P+9a9/0dLSwpIlS865b15eHiUlJbS3nz09Y7VaiY2N7XZTGAwtFXfgA3B26GvLQKgvhSLPHCrtPQGTM21kxEXQYney9uBJnYzzDR/sFivaqyel6WyJwqcMGQGpk8RgXaPXD2oLlvGLug3O1Wp7jB7h3XishoY2B4nRYYNmcO7p6OosJSUlMXbs2F5vYWFhzJ49m7q6OrZv3+597ieffILL5SIvL++cr/P888+zaNEikpLOLWSXn59PfHw8VqsqIpWaobMhKgna6uD4Or2tOX/2vyPusy4AW6b3YZPJ5F0BauFzI3Kysd2r63J1rnKWpGPiV8X93rf0tWMgONrhoEfkdvzibr/SIrxbC2sNnYpb1SUFZxmEKTgwSM3SuHHjWLhwIcuWLWPLli188cUX3HXXXdx4442kp6cDUFpaytixY9myZUu35x45coR169Zx2223nXHc999/n+eee469e/dy5MgR/vKXv/Doo4/ygx/8ICDvS6EjZktnKm6fgS/U2peM9qXThas8kZg1BcZNxa3aV4HLDZOz4shKUCk46ZjwFXF/4gvjjiA6sgbsjRCbARnTu/0qPS6CmdnxuN2dEVKjYXe4+M8eURs8mBcshpHBfeWVV7jrrru4/PLLMZvNXHfddTz11FPe33d0dHDw4EFaWroLDb7wwgtkZmZyxRVXnHHM0NBQnn76aX70ox/hdrsZOXIkTzzxBMuWLfO5/S6XC7tdLkVlwzP2OjiwGgp3QFMDhAxszlFoaCgWi8VHxvWBUyfEnDtM3VJwGlOz4ki3hVNW38bagycN2e67cpdoK//yIL5IS018tnAwSreLVFbe2bXzgpY9HpHj8YtFPeRpXDM5na2Fp3h/dzm3XZgTWNt8wLpDJ6lv7SA5xkpezhC9zdENwzhLCQkJvPrqq2f9fXZ2do8t/48++iiPPvpoj89ZuHAhCxcu9JmNZ8Nut3P8+HFcLpffX0vRD9wJcOEfweWAY4chdOCRC02zy2QKQKh639viPnueaMU+DZPJxDWT0/nrumO8m19qOGepqqHN2wV3Za6xbFf0g4nXCWdp31vGc5baGztHJ026vsddrpyYxi/f28eu4jqKaloM16TwrrZgmZQ+aFNwYCBnyai43W7Ky8uxWCxkZWVh7mHlodCRxihorYGwWIjLPPf+Z8HtdtPS0uJtREhLC0AkZO+/xX0PKTiNa6dk8Nd1x1hTUEV9awe2iFD/2+Uj3ttVhtsN04bGkRlvrC8YRT8Yvxg+/JloVKgv6VZ7F/QUrARHKwwZBWlTetwlKcbKnBGJfH6kmvd3l/H9S0cG1sYB0NzuYLVnzNC1U9J1tkZflLPkZxwOBy0tLaSnpxMZqS74QYc5CTpqwdUEYSFgPv9TIiIiAoCqqiqSk5P9m5Kr3A8Vu4W9485MwWmMS4thTEoMBysb+e+ecm6cNdR/NvmYt3aUAvCVaQb68lT0H1sGDJsr6pZ2vwkX3qO3RX1nz5viftLXu3XBnc41k9P4/Eg17+WXceclIwITefYBHxdU0tbhIntIJJMybXqboysqzOFnnE5RWBsWNrB6GIWfCI2AkHDADW31Az6c5hB3dPhZjmDXa+J+1AKIOnsdgclkYvHUDADe3lnqX5t8SEF5A/vLGwi1mLhGSQbIz+Qbxf2u1+AcExSChsbKTlHbidf1uuvCiWmEhZg5WNnIvjLjTHx4N1+k4BZNyTCMg+cvlLMUIAb7By1oMZkgQgjH0VLrg8MF4P/scooVOMCUb5xz90We8Pnm47WU1hmjfVlz7C4fm0JcpFpoSM/4xRASAdWHoGyH3tb0jX1vgdsFGTOEZlQv2CJCvaN6/rW9JBDWDZiqxjY+OyQ02hZNHtwpOFDOkkLR6SzZm4wxK+7Yp9BUIeweteCcu2fERZA3XAjJvWOA6JLT5fba+ZVpGTpbowgI4bGdCvT5r+lrS19wu2HnK2J70tf79JSvTRfp5HfyS2l3BL+Ux9s7SnG63EwdGsfI5OhzP0FylLOkUIRYIcxzMfBBdMnvaF8mE7/WZ7mD6zx1P//cVnzOQdF688WRaqoa24mLDOXSMcnnfoJCDrRU3N5/Bf+ipTwfKveAxQq5PXfBnc6Fo5JIibVS19LBpweCe8C12+3mzW1ijNfXZ6jZp6CcJcVZcLvd3H777SQkJGAymcjPz6empobk5GQKCwv7dAy73U52djbbtm3zm53Z2dk8+eST/X7eL37xC26/vUubcqSn7qe1tseaiWeeeYZrrrnmPK30IW31cGCl2O5DCk7j6klpRIVZKKxpYdOx4HYI3/BcpK+ZlE5YiLpEDRpyLhUDrltPwaEP9bamd3b8Q9yP+zJE9m38h8XcWT8Y7Km4HUV1HD3ZTHiomS+rmkFAOUuKs7Bq1SpefPFFVq5cSXl5ORMnTuTXv/411157LdnZ2X06RlhYGD/5yU+47777/Gbn1q1buzs9faCiooI//vGP/PznP+98MDwOTBZw2oV2yml85zvfYceOHaxfv36AFg+Q3W+Cow2SxkL6tD4/Lcoa4q1demNrkb+sGzDVTe18tE+0Kt84S61oBxVmS2d0afuLuprSKx2tsOdfYnvqt/r11K95IryfHjxJVUObry3zGf/0LFiuyk0jJtw4ciP+RDlLAcbtdtNid+hy60/65ejRo6SlpTFnzhxSU1Ox2+08//zzLF26tF/v96abbuLzzz9n3759/f1T9YmkpKR+SzI899xzzJkzh2HDhnU+aDZDpFboXXPGc8LCwvjmN7/ZTTU+4LjdsO0FsT3jO722KvfEjTOFbMB/9lZQ3xKcw4P/ua2EDqebyVlxTEgf3K3Kg5Jp3xb3R9dA7XF9bTkb+9+D9nqIGwrDL+7XU0elxDBtaBxOl5s3thb7ycCB0WJ38L5HiFKl4DpROksBprXDyfgH9Qkx7394AZFh5/6X33LLLaxYsQIQ3V3Dhg3j8ccfx2q1csEFF3j3e/jhh3nmmWfYs2cPQ4aINNbVV19NS0sLa9aswWw2Ex8fz9y5c3n99dd55JFH+m2z2+3mV7/6FS+88AKVlZUMGTKEr33ta16nJTs7m7vvvpu7777ba++zzz7LBx98wIcffkhGRga///3vWbRokfeYr7/+OnfccYf355MnT5Kbm8v/+/4d/GzpImirZ8P6dVxy+Xz++9//cvnllwNwzTXX8KUvfYnW1lavplJAKd4MVftF19CkG/r99EmZNsamxnCgopF38kv59pxs39s4AFwuN697ol43GUgPSuFDEobDiMuFs7T9RfjSr/S26Ex2vCTup36rx/Em5+Jbs4exo6iO17YUccclIwixBFfM4p2dZTTbnQwbEultDFGoyJKiB/74xz/y8MMPk5mZSXl5OVu3bmX9+vVMn959SOTPf/5zsrOzvUOKn376aTZs2MCKFSu6KZXPmjWrW/rqlVdeITo6utebtv+///1v/vCHP/DXv/6Vw4cP884775Cbm9ur/b/61a/4+te/zu7du7nqqqu46aabqK0VdTq1tbXs37+fGTNmePdPSkrihRde4JeP/Jpt+47S2NTEt5Ys8c4i1JgxYwYOh4PNmzef5192gGx9XtznXgcRcf1+uslk4hseJ+SVzSeCrtB7w9EaTtS0EGMN4cuTVZ3EoGXGd8T9zn8EX6F3xV448blI2U+56bwOceXENOIjQymrb+OTICv0drvdvLSxEIBvXTBMSd50QUWWAkxEqIX9D5+73dtfr90XbDYbMTExWCwWUlPFTK4TJ06Qnt5da8NisfDyyy8zZcoU7r//fp566imee+45hg7tHhVIT0/nxIkT3p8XLVpEXl5erzZkZIhCyKKiIlJTU5k/fz6hoaEMHTqUWbNm9frcW265hW98QxQ/P/roozz11FNs2bKFhQsXUlRUhNvtPuO9XHXVVSxbtoybvn8/MyaOJioijMce/XW3fSIjI7HZbN3eS8BoroH974jtGf1LhXblK9My+O2qAxyqbGLD0Rrmjkz0jX0+4JXN4u+6eGpGnyKgCkkZvRBi0qGxDAreh9yv6W1RJ1v+Ku7HXSOUx8+D8FALX5+ZxV8/O8bLm4u4YkLwzD3ccryWAxWNRIRauH66SsF1RV2RAozJZDLkF0Frayvh4eFnPJ6Tk8Pjjz/Od7/7XW644Qa++c1vnrFPREQELS0t3p9jYmKIiYnp0+tef/31PPnkk+Tk5LBw4UKuuuoqrrnmGkJCzv43nDRpknc7KiqK2NhY78y21lYhytjTe3n88ceZOHEV/1y5mu3/fQWruw3onm47/b0EjB0rRPF52hTI6Hth9+nEhody/fRMVmw8wQufHw8aZ6m4toUPPYXdN12gUnCDGksITP82rH0MtvwteJylltpOMdi87w3oUDfNGsbf1h1j3aGTFFY3k50Y5QMDB84KT1Rp8dQMbJGqsLsrKg2n6BOJiYmcOnWqx9+tW7cOi8VCYWEhDofjjN/X1taSlJTk/bk/abisrCwOHjzI//3f/xEREcGdd97JRRdd1Os4kdDQ7ie5yWTC5XJ53wfQ43s5evQoZWVluFxuCovLoPnkOd9LQHC0w2bPijbvuwM+nFar9MnBKo5XNw/4eL7ghS+O43LDhaMSGZsaq7c5Cr2ZfgtYwkSdXvEWva0R7HhJdKKmToKhF5x7/14YOiSSi0eL68jfvwiOQvby+lY+3FcJwJLZw86x9+BDOUuKPjF16lT2799/xuNvvPEGb731FmvXrqWoqKjHIu69e/cydepU78+LFi0iPz+/11vXmqKIiAiuueYannrqKdauXcvGjRvZs2fPeb2PESNGEBsbe8Z7sdvt3Hzzzdxwww088qtfctu9j1BVVgz2zijS0aNHaWtr6/ZeAsKefwnF7pg0IUQ5QHKSorlsbDJuN6zYUDhw+wZIfWsHb3o6g267MEdnaxRBQUxqpzL2Bh07UDWcDtj6nNjO+16/O1F74nbPZ/2NbcXUNtsHfLyBsmLDCZwuN7OGJzAuTS1YTkc5S4o+sWDBAvbt29ctIlNSUsIdd9zBb3/7W+bNm8ff//53Hn30UTZt2tTtuevXr+eKK67w/hwTE8PIkSN7vWndZi+++CLPP/88e/fu5dixY7z88stERER0b/vvB2azmfnz5/P55593e/znP/859fX1PPXUU9y3/GeMHpnDd+75JTRVdnsfOTk5jBjR+xwon+J2w4Y/ie287/VZsftc3Do3GxB6KnUt+l6oX99SRLPdyeiUaC4aFRxpQUUQMPsucV+wEmqO6mvL3n9BfTFEJp5zaG5fmT1iCLkZNto6XN6iar2ob+3g5U2iZnCZWrD0iHKWFH0iNzeXadOm8eabImfvdru55ZZbmDVrFnfdJS5qCxYs4I477uDmm2+mqakJgI0bN1JfX8/XvnZ+EZG4uDieffZZ5s6dy6RJk/j44495//33vVIF58Ntt93G66+/7k3NrV27lieffJJ//OMfxMbGYjab+cdL/2D9lnz+8tdnoUOIx7322mssW7bsvF/3vDjyMZwsEONYpt/is8POG5nI2NQYmu1OXvii0GfH7S92h4sXPdGt2+blqO4bRSfJ42DUFYAbNj6tnx0uF6z/vdiefSeEnlnveD6YTCa+e7FwTFZsKKTVrt+8uH9sLKSp3cGYlBguH6tGDPWEyR1s/cMGpKGhAZvNRn19PbGx3cOXbW1tHD9+nOHDh/dYVGwkPvjgA+6991727t3bTRqgN2644QYmT57Mz372Mz9b13fcbjd5eXn86Ec/8nbN9UjtMTFeJCKBfWVNXHbZZRw6dAib7exiiT79f7vd8PeroGiDWGUv+PW5n9MP/rOnnDtf2UFMeAhf3H8ZsToo9b686QQPvLOX5Bgr6356KeF97NhUDBKOr4cVX4aQcPjhLpGeCzT73oF/fhvCbXD3XjH010c4nC4u+/1nFNW28MtrxnPL3OE+O3ZfabE7mPfbT6lttvPHG6dw7ZTBNby6t+/vrqjIkqLPXH311dx+++2UlvZtcr3dbic3N5cf/ehHfrasf5hMJv72t7/1WIzejegUcd9aS3nJCV566aVeHSWfc/QT4ShZrDD7+z4//MIJqYxKjqaxzcEKHaJLbR1Onv70CADfv3SkcpQUZ5I9D7LyRGG1Ft0JJG43rHtcbOd9z6eOEkCIxcyyC4WD9H9rj9LWEfjo0utbRM3U0IRIrs5V+mZnQzlLin5x9913k5XVN/2NsLAwHnjgAX3Urs/BlClT+Na3zjHXKSwKrELiYH7eRBYsCKA+ltsNn3iK5WfeBrHpve9/HpjNJu66bCQAz31+nKb2cziPPuaNrcWU17eRZgvnhplK00XRAyYTXPaA2N72d6gL8FzDg/+Byj0QGjVguYCz8fWZWWTERVDV2B7whoumdgf/t1YsWL57cU7QqYkHE+ovo1D0RoxnpdVSIwZoBoqD/4GynRAaCfP8F5n78qR0cpKiqG/t4G+fBa6IVkWVFH1m+EXi5uqAz34XuNd1dsDqh8R23u0Q6Z/RH9YQC3fPHwXAXz47SkNb4OY2PrvuGNVNdrKHRCoRynOgnCWFojfCokStAkBDWWBe0+mAT/5HbOd9D6L9p+tkMZu494oxAPxt/THK6gLjED63/hhVje1kxEWoYZ2Kc3PZL8R9/qtw8lBgXnPHCqg5DJFD/LpgAfjqtExGJkdT19LBc+uO+fW1NKoa23h2vXitexeMJSxEuQO9of46CsW5iE0HTNDeAG0N/n+9bc+LgbnhcTDnB35/uYUTU5k1PIG2Dhe/W3XA769XVtfK05+KKNZPF45RF2nFucmaBaOvBLcT/vtTkab2J20N8OljYvuS5Z0LJj9hMZv4yRWjAXh2/XFKTvl/SsAfPz5Mi93J5Kw4rsoNnpErwYq6SikU5yIkHKI8+j8NpeB2+e+1mqo6o0qXP+i30H9XTCYTv7h6PCYTvJNfxs6inpXafcWj/ymgtcPJrOwEFk32fS2WQlIWPiqaHY59Cvvf9e9rrftfaKmGIaN8KtnRGwsmpJI3PIHWDie/fO9MAWBfsqeknte2iPqvn105Vkl29AHlLCkUfSE6VUwad7QJh8ZfrH5IRLDSpgTsIg2Qm2njq1MzAVj+1h7sDv84hBuP1rBydzlmE/xy0QR1kVb0nYQcmHe32P7wZ9De5J/XKd0BG/8stq/4H7AERlLDZDLxP4snEmI28XFBJR95ZiX6mg6ni5/+ezcuN1wzOZ28nPPXrBtMKGdJoegLlhCwCWeCxgr/FHsfWQO7XgVMcPUTYA5s0fPyq8aSEBXGgYpG/vzJYZ8fv7Gtg5/+excAN+UNY3y6Gqmg6CfzfgRxQ0WEd82vfH98hx3evUtEjydeB2MW+v41emFUSgzLLhJClb96fz+Nfij2fnb9MQrKG4iLDOWha8b7/PiyopwlhaKvRMSD1Qa4RQuzL+smmk7C257W5Jm3QeZ03x27jyRGW3nk2okAPL32KHtK6n16/F++t5/i2lYy4yP46cIxPj22YpAQGgFX/0Fsb/kbHFzl2+Ov/z1U7RNF3VcGsPOuC//vslFkxkdQWtfKz9/eiy91ow9VNvLkx2Ih9Iurx5MYbfXZsWVHOUuKfnHJJZdw9913A9DS0sJ1111HbGwsJpOJuro6AA4ePEhqaiqNjY19OmZ1dTXJycmUlJT0257CwkJMJhP5+fn9fm6/MZkgLlOk4zpaxOrWF7jd8O6d0FwFSePgijOHEQeKqyel8eVJaThdbn74+k7qW32zsv1gdzn/3lGC2QR/uGEKMTqohSskYdR8yLtDbL97p4j0+oIja2Cdx0G68neddYoBJiLMwh9vnILFbOK9XWW8ua3YJ8dtaOvge//Yjt3h4uLRSXx12uBS6h4oyllSnDcrVqxg/fr1bNiwgfLycq+69fLly/nBD35ATExMn46TmJjIkiVLeOihh3r8/a233soDDzzgM7sHhCVMpAEAmk8K/aWBsu5xOPyRKCT/2gti9awjD187kXRbOMeqm/nBaztxOAdWv7S3tJ6f/kuk3+68ZCQzs/1ftK6QnC/9ClJzxfn3z1u98xvPm9pj8K/viPTb1Jt9Niz3fJk+LIGfeCQ9HnpvH/vKBhbldbvd/OTNXRyrbibdFs4TX5+s6gX7iXKWAo3bDfZmfW4+brc9evQo48aNY+LEiaSmpmIymSgqKmLlypXccsst/TrWrbfeyiuvvEJtbW23x51OJytXrmTRokU+tHyARMSJgm+AuuKBFZpufxE+9XS/LfwNpOhfQ5AQFcbflswgItTCukMn+Z8PCs47FVByqoVbX9xKs93JnBFD+KFHfE+hGBAhVrjuBbDGipFA/14q9MnOh5ZaeP0maKuDzJmiXjAIHInvXpTDRaOTaOtw8e0XtnK8uvm8juN2u/nNqgN8tL+SMIuZv9w8nSEq/dZvQvQ2YNDR0QKP6tQu/bMyIbLYR5qbm7njjjt46623iImJ4Sc/+Yn3d5dccgmfffYZILo4Lr74YtauXcubb77J5MmTycjoDPF+5zvfYdu2bWzduhWr1YrdbicvL4/c3FxeeuklACZMmEB6ejpvv/02S5cu9T53w4YNhIaGMnPmTAC2bNnCd7/7XQoKCpg4cSI///nPu9n88MMP88wzz7Bnzx6GDBFdHldffTUtLS2sWbOmzwOAz0lMqijybq+H2qOiU8fat0ial71vwUqP2N2FP4EZt/rGNh8wMcPGE1+fzB2v7ODFDYWYTKLGwWzu+5dIyakWlrywhZON7YxNjeGZb00nVI1TUPiKpNHwjdfgH1+FAyvh/R/CNX8UzRh9pakKXlosdM2iU+Dr/xCOWBBgNpv40zem8o2/bWJ/eQM3P7eZN783m4y4vkee3W43v/nvAf7qEbr8n8UTmZwV5yeL5cYwV65f//rXzJkzh8jISOLi4vr0HLfbzYMPPkhaWhoRERHMnz+fw4e7d/nU1tZy0003ERsbS1xcHEuXLqWpyU8tqQbj3nvv5bPPPuPdd9/lo48+Yu3atezYsQOAt956i2XLljF79mzKy8t56623AFi/fj0zZszodpynnnqK5uZm7r//fgB+/vOfU1dXx5///Odu+82aNYv169d3e+y9997jmmuuwWQy0dTUxJe//GXGjx/P9u3b+eUvf9nNgdOOnZ2dzW233QbA008/zYYNG1ixYoXvHCUQK8/4YcJBcrug5ii09lGfyOWCtb+Ff90qnjvt253zr4KIK3PT+KWnW+bvXxRyz5v5fR70ubPoFIuf3sCxk82k2cL5+60ziVV1Sgpfkz0PvvY8mMyQ/zK88jURKeoL1Yfh71eKgu7oVFjyHsQG1yBZW0QoLy2dRU5SFKV1rVz75y/YdKxvqf9Wu5Ofvb3H6yg9fO0Evq5mMJ43hoks2e12rr/+embPns3zzz/fp+f87ne/46mnnmLFihUMHz6cX/ziFyxYsID9+/cTHh4OwE033UR5eTmrV6+mo6ODW2+9ldtvv51XX33VP28kNFJEePQgNLLPuzY1NfH888/z8ssvc/nllwOiRikzU7TPJyQkEBkZSVhYGKmpneqvJ06cOMNZio6O5uWXX+biiy8mJiaGJ598kk8//ZTY2O6t4+np6ezcubPbY++++y5/+IPofnn11VdxuVw8//zzhIeHM2HCBEpKSrjjjju8+1ssFl5++WWmTJnC/fffz1NPPcVzzz3H0KFD+/ze+4zZIiJKtYUiwnSqEMyx4OrFoag6AKt/IWqUAGZ9FxY8GhRh/564Ze5wbJGh3PvP3byTX8auknp+vXgic0b2XPza3O7gr+uO8cxnR7E7XIxLi+X5b88gzRZ8w5QVkjDuGrj+RdFNeuxTePZScU6Nuarn88rlhI1Pw6e/FrpptixY8i4MGRFw0/tCYrSVV27L49a/b+VARSM3PbeZH1w2kqXzhp+1UWJn0Sl+8s9dHD0pUnePLJ7Ity4YFkizpcMwztKvfiU0NV588cU+7e92u3nyySd54IEHuPbaawF46aWXSElJ4Z133uHGG2+koKCAVatWsXXrVu8X/J/+9CeuuuoqHn/8cdLT/ZAuM5n6lQrTi6NHj3rTZRoJCQmMGdN7y3dra6vXEe3K7Nmz+clPfsIjjzzCfffdx7x5887YJyIigpaWTpn/goICysrKvM5aQUEBkyZN6nb82bNnn3GcnJwcHn/8cb773e9yww038M1vfvPcb/h8MZkhIVt05DRVgr0eGqvhoxUw6iIxiNcSCuW74OgnsO8dwC0Kxb/8B1FMGuR8ZWomQ6Ks/OSfuzhe3cw3n9vMzOx4rs5NIzfThjXEQlVjGxuO1PBOfhnVTe0AzB+Xwh9vnEKU1TCXGYVRGX+tWLi8/k2xaHn9m5A+FSZ8VdQhhUWKYvDj62D3m52drCMug2uf9ow0Cl7SbBG8fedcfv72Ht7aWcqTHx/mhc+Pc+OsoUwbGk9OUhSnmu0cq27mX9tL2H5CRLmTY6w8fv1kLhrtv/mSgwVpr2LHjx+noqKC+fPnex+z2Wzk5eWxceNGbrzxRjZu3EhcXFy3SMj8+fMxm81s3ryZr3zlKz0eu729nfb2du/PDQ0BmBdmEBITEzl16sx0lMvl4osvvsBisXDkyJEen1tbW0tSUudJ/d577/GlL32pR+frXKxbtw6LxUJhYSEOh4OQED9+1E1mcbENt8HJE6KQ/tB/Ycffet5/3CK49OeQPNZ/NvmYi0Yn8fGPL+bxDw/yj00n2Fp4iq2FPacdhw2J5P6FY1k4MVV13CgCR2oufHc9bPgTbPoLlO0Ut56IiIcvPSIWKwb5jEaEWfj9/2/vXmOiuvY2gD8DAyPIiCAXoSIC6lGPF1CEE6EFq3E0DT1EA72IFtKgELBYbWqtnmJaK7XY1KBVwbbEGFtNY414qVaxYmK1qIjh4uVYtVAoAw12oAiKzDofeJ3jFNnieYW1geeXzIe9Zs/wuAaZ/6y9Zq3YSYj4mzuy8v+Nn+uakNPJprtaGw1eDPTGv14YB5eB9j2ctG/qs8VSTU372huenp5W7Z6enpb7ampq4OHhYXW/VquFq6ur5ZxHycjIsIx09VUBAQGws7PDTz/9ZLmEdfv2bVy7dg0RERGdPi4oKAjl5R33NcrMzMSVK1dQUFAAg8GA3NxcJCRYT2guLS1FZGSk5Xj//v1YtGiR5Xjs2LHYuXMnWlpaLAXU2bNnO/ysPXv24Ntvv8XJkycRGxuLDz74oGdeL/uBgKsfUN8KjI8Bbh1vn8d0rwnwGAcMCwbGzwW8JnV/lm4waIAd3v/neCRFBOC70hocLauBsaEFzffaMFCnRaifK8JGusHw96HcHJfkcBgMzPgX8I9koPgroPKn9u1LhLl9n0VXf2BiLDB6tmomcj8JjUaDfwY+g6iJ3vi+vAbHL9fi8m8NqKi/AzcnHYYOGoCwkUMQO9UHHvon/5BJCoREK1asEAAUb5cvX7Z6TG5urnB2dn7sc58+fVoAENXV1VbtMTExIjY2VgghxIcffihGjx7d4bHu7u5iy5YtnT53S0uLMJlMlltlZaUAIEwmU4dzm5ubRXl5uWhubn5sZrVJSkoSvr6+Ij8/X5SUlIgXX3xRODk5ibS0NCGEEGlpaSIiIsLqMXl5ecLDw0Pcv3/f0lZUVCTs7e1FXl6eEEKI7Oxsodfrxc8//2w5p6mpSTg4OIhTp04JIYQwGo3Czs5O1NXVWc5pbGwUbm5uIi4uTpSVlYlDhw6JkSNHCgDi4sWLQgghKisrhYuLi8jKyhJCCHHkyBGh1WrFmTNnnnb3PFJvfr2JiPobk8nU6fv3w6SOLC1fvvyx6/H4+/v/T8/9YNKx0WiEl9d/v+FgNBoRGBhoOae21npT1Pv376O+vt5q0vJf6XQ66HS971PJk8rMzMSff/6JqKgo6PV6LF++HCaT8uJoc+bMgVarxfHjx2EwGNDS0oK4uDjEx8cjKioKALBo0SIcOnQICxYssFwu279/P4YPH45nn30WAHDgwAGEhITAze2/E4mdnJxw4MABJCUlISgoCOPGjcP69esxb177AnJCCMTHxyMkJASpqakAAIPBgOTkZMTFxaG4uBhOTk7d0VVERNSX9Uzt9vR0dWTJbDaLoUOHig0bNljaTCaT0Ol04uuvvxZCCFFeXi4AiPPnz1vOOXr0qNBoNKKqqqrLmZQq0/440rB582Yxa9asJ3pMaGio2LVrl+U4KipKrF+//mlH63b98fUmIuqtujqy1GsmFlRUVKC4uBgVFRVoa2tDcXExiouLrdZEGjNmDPbt2weg/dru0qVLsXbtWuTl5aGkpAQLFy6Et7c3oqOjAbTPgZk9ezYSExNRWFiI06dPIzU1FS+//HL3fBOun1i8eDGee+65J9obbu7cuXjllVcsbeHh4VbHREREsmiEeMp7YHST+Ph47Nixo0P7Dz/8YJkUrNFokJuba7m0J4RAeno6cnJy8McffyA8PBxbtmzB6NGjLY+vr69HamoqDhw4ABsbG8ybNw9ZWVlPdLmmoaEBzs7OMJlMHdYOamlpwc2bN+Hn5/c/fauLehe+3kREvYfS+/fDek2xpGYslugBvt5ERL1HV4ulXnMZrrdjTdo/8HUmIup7WCx1M1tbWwDt27VQ3/dgBXI7O+6DRkTUV/TZRSnVQqvVwtHREXV1dbCzs3u6m7mSagghcOfOHdTW1mLw4MGWIpmIiHo/FkvdTKPRwMvLCzdv3sQvv/wiOw51s8G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      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return torch.sin(x)\n",
    "\n",
    "def df(x):\n",
    "    x.requires_grad_()\n",
    "    y = f(x)\n",
    "    y.backward(torch.ones_like(y))\n",
    "    return x.grad\n",
    "\n",
    "x = torch.linspace(-2*torch.pi, 2*torch.pi, 200)\n",
    "\n",
    "y1 = f(x)\n",
    "y2 = df(x)\n",
    "\n",
    "plt.plot(x.detach().numpy(), y1.detach().numpy(), label='f(x)=sin(x)')\n",
    "plt.plot(x.detach().numpy(), y2.detach().numpy(), label='df(x)/dx')\n",
    "plt.legend()\n",
    "plt.xlabel(\"X\")\n",
    "plt.ylabel(\"Y\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2.6 概率"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.6.1 \n",
    "\n",
    "进行$m=500$组实验，每组抽取$n=10$个样本。改变$m$和$n$，观察和分析实验结果。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：** 抽取的样本越少达到需要的进行实验的组"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "import torch\n",
    "from torch.distributions import multinomial\n",
    "from d2l import torch as d2l"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "fair_probs = torch.ones([6]) / 6\n",
    "def experiment(n=10,m=500,legend=1):\n",
    "    counts = multinomial.Multinomial(n, fair_probs).sample((m,))\n",
    "    cum_counts = counts.cumsum(dim=0)\n",
    "    estimates = cum_counts / cum_counts.sum(dim=1, keepdims=True)\n",
    "\n",
    "    d2l.set_figsize((6, 4.5))\n",
    "    for i in range(6):\n",
    "        d2l.plt.plot(estimates[:, i].numpy(),\n",
    "                 label=(\"P(die=\" + str(i + 1) + \")\"))\n",
    "    d2l.plt.title(f'n={n},m={m}')\n",
    "    d2l.plt.axhline(y=0.167, color='black', linestyle='dashed')\n",
    "    d2l.plt.gca().set_xlabel('Groups of experiments')\n",
    "    d2l.plt.gca().set_ylabel('Estimated probability')\n",
    "    if legend==1:\n",
    "        d2l.plt.legend();   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
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    "d2l.plt.figure(figsize=(10, 10)) \n",
    "d2l.plt.subplot(231)\n",
    "experiment()\n",
    "d2l.plt.subplot(232)\n",
    "experiment(1,500,0)\n",
    "d2l.plt.subplot(233)\n",
    "experiment(100,500,0)\n",
    "\n",
    "d2l.plt.subplot(234)\n",
    "experiment(10,50,0)\n",
    "d2l.plt.subplot(235)\n",
    "experiment(1,50,0)\n",
    "d2l.plt.subplot(236)\n",
    "experiment(100,50,0)\n",
    "d2l.plt.subplots_adjust(wspace =0.5, hspace =0.5)"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.6.2 \n",
    "\n",
    "给定两个概率为$P(\\mathcal{A})$和$P(\\mathcal{B})$的事件，计算$P(\\mathcal{A} \\cup \\mathcal{B})$和$P(\\mathcal{A} \\cap \\mathcal{B})$的上限和下限。（提示：使用[友元图](https://en.wikipedia.org/wiki/Venn_diagram)来展示这些情况。)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;$P(\\mathcal{A} \\cap \\mathcal{B})$的上限是 $ \\min(P(\\mathcal{A}), P(\\mathcal{B}))$，因为两个事件的交集的概率的上限不能超过两个事件中概率较小的那个。$P(\\mathcal{A} \\cap \\mathcal{B})$的下限是 $0$，因为两个事件可以互斥。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "所以\n",
    "$$\n",
    "0 \\leqslant P(\\mathcal{A} \\cap \\mathcal{B}) \\leqslant \\min (P(\\mathcal{A}), P(\\mathcal{B}))\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;由于$P(\\mathcal{A} \\cup \\mathcal{B})=P(\\mathcal{A})+P( \\mathcal{B})-P(\\mathcal{A} \\cap \\mathcal{B})$，根据$P(\\mathcal{A} \\cap \\mathcal{B})$的上下限，可以得到\n",
    "$P(\\mathcal{A} \\cup \\mathcal{B})$ 的上下限：\n",
    "\n",
    "$$\n",
    "\\max(P(\\mathcal{A}), P(\\mathcal{B})) \\leqslant P(\\mathcal{A} \\cup \\mathcal{B}) \\leqslant P(\\mathcal{A}) + P(\\mathcal{B})\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.6.3\n",
    "\n",
    "假设我们有一系列随机变量，例如$A$、$B$和$C$，其中$B$只依赖于$A$，而$C$只依赖于$B$，能简化联合概率$P(A, B, C)$吗？（提示：这是一个[马尔可夫链](https://en.wikipedia.org/wiki/Markov_chain)。)"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：**"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;根据题意，由于$B$只依赖于$A$，可知\n",
    "$$\n",
    "P(B | A, C) = P(B | A)\n",
    "$$\n",
    "由于$C$只依赖于$B$，可知\n",
    "$$\n",
    "P(C | A, B) = P(C | B)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "可得：\n",
    "$$\n",
    "\\begin{aligned}\n",
    "P(A, B, C)\n",
    "&= P(C|A, B) P(A, B) \\\\\n",
    "&= P(C|A, B) P(B|A) P(A) \\\\ \n",
    "&= P(C|B) P(B|A) P(A)\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "则：\n",
    "$$\n",
    "P(A, B, C) = P(C | B) P(B | A) P(A)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 练习2.6.4\n",
    "\n",
    "在2.6.2节中，第一个测试更准确。为什么不运行第一个测试两次，而是同时运行第一个和第二个测试？"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答：** "
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;根据题意，如果同时运行第一个测试两次且他们互不影响。\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "& P(D_1 = 1, D_1 = 1 \\mid H = 1) \\\\ \n",
    "=& ( P(D_1 = 1 \\mid H = 1))^2 \\\\\n",
    "=& 1 \n",
    "\\end{aligned}\n",
    "$$"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{aligned}\n",
    "&P(D_1 = 1, D_1 = 1) \\\\\n",
    "=& P(D_1 = 1, D_1 = 1, H = 0) + P(D_1 = 1, D_1 = 1, H = 1) \\\\\n",
    "=& P(D_1 = 1, D_1 = 1 \\mid H = 0) P(H=0) + P(D_1 = 1, D_1 = 1 \\mid H = 1) P(H=1) \\\\\n",
    "=& (P(D_1 = 1 \\mid H = 0))^2 P(H=0) +( P(D_1 = 1 \\mid H = 1))^2 P(H=1)\\\\\n",
    "=& (0.01)^2 \\times 0.9985 + 1 \\times 0.0015 \\\\\n",
    "=& 0.00159985 \\\\\n",
    "\\end{aligned}\n",
    "$$"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{aligned}\n",
    "&P( H = 1 \\mid D_1 = 1, D_1 = 1) \\\\\n",
    "=& \\frac{P(D_1 = 1, D_1 = 1 \\mid H = 1) P(H=1)}{P(D_1 = 1, D_1 = 1)} \\\\\n",
    "=& \\frac{1 \\times 0.0015}{0.00159985)} \\\\\n",
    "=& 0.9375878988655186 \\\\\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "&emsp;&emsp;根据得到的结果，高于同时运行第一个和第二个测试的概率值 ，但现实中我们不会让病人进行两次同种检验，另外如果进行相继两次的第一种测试，得到都为阳性的结果患者得病的概率为$1-(1-0.1307)^2 = 0.24431751000000002$"
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